The von Mises-Fisher distribution has two parameters: the mean $\mu \in \mathbb{R}^p$ and concentration $\kappa \geq 0$, where $\mu$ is constrained to have unit norm.
Why not instead define the distribution for one unconstrained parameter i.e. $y := \kappa \mu \in \mathbb{R}^p$?
What is gained (conceptually or computationally or statistically) by thinking in terms of two parameters rather than one?
The von Mises-Fisher distribution is a distribution on the surface of a sphere. To make it as easy to visualize, think of a circle. The vM-F distribution has two parameters: the mean direction in which points are distributed on the circle, and how concentrated they are around the point on the circle in that mean direction. You can think of this as analogous to the mean and standard deviation of a Normal distribution, or the median and scale parameter of a Cauchy distribution, or... In higher dimensions, the parameter interpretations are the same.
In the higher dimensions, an alternative way of thinking of it is as a point on the surface of a sphere which is the center of the distribution and how "spread out" the distribution is around that point. If κ=0, the spread is such that the points are uniformly distributed on the surface of the sphere, but as κ gets large, the distribution concentrates around the center point. You can think of these two features of the dist'n independently of each other, and it is often helpful to do so.
As @Scortchi - Reinstate Monica points out in response to a comment below:
"... there are plenty of applications in which $\mu$ is the parameter of interest & $\kappa$ a nuisance parameter - testing hypotheses about $\mu$, making point & interval estimates of $\mu$, are the aims of the study. It's hard to think of an application where the components ($\kappa \mu_1,\kappa \mu_2, \dots, \kappa \mu_p$) would be of separate interest."
In geometry we represent directions with a unit norm vector, so $\mu$ represents the direction. That's why its norm is $||\mu||=1.$ The other parameter is similar to variance in that it represents how directions are concentrated around $\mu.$
You can pack $\kappa$ into $\vec\mu$ as its magnitude $\vec\mu'=\kappa\vec\mu$, then $||\vec\mu'||\ne 1$, but this will create uninterpretable artifacts: concentration components in different directions $\kappa_x,\kappa_y,\kappa_z$. This is different from a technically similar trick in physics: packing a magnitude $a$ of acceleration into a length of a vector $\vec a$ with $a=||\vec a||\ne 1$. The acceleration components $a_x,a_y,a_z$ have a clear physical interpretation here.
The distribution in question, although technically operates in nD space, concerns with directions on (n-1)D sphere. So, trying to use up that extra dimension may look like a clever way to parsimony, but it in fact doesn't correspond to the mature of the phenomenon modeled.
Why does the von Mises-Fisher distribution need two parameters?
We do not necessarily need two sets of parameters: location and scale. You can combine them into one set or in another alternative parameterization. But, nobody has found it useful to describe the distribution that way (or at least not enough people such that it has become mainstream).
The von Mises-Fisher distribution can be parameterized in terms of the location on a sphere. But this can also be parameterized in terms of spherical coordinates.
For instance, we have for the von Mises distribution (the 2d special case) the two descriptions
in terms of the angle $\theta$
$$f(\theta; \mu_{\theta},\kappa) = \frac{1}{2\pi I_0 (\kappa)} e^{\kappa \cos(\theta-\mu_\theta)}$$
in terms of the point $\vec{x} = \lbrace x_1, x_2 \rbrace$ on a circle of unit length
$$f(\vec{x}; \vec{\mu},\kappa) = \frac{1}{2\pi I_0 (\kappa)} e^{\kappa \vec{x} \cdot\vec{\mu} }$$
Why does the von Mises-Fisher distribution need two parameters?
The term 'parameter' here is a bit confusing. The distribution actually needs $n$ parameters. What you are referring to is the split-up of those $n$ parameters into a set of $n-1$ parameters relating to the location and a parameter relating to the scale. Those two sets, instead of combining them into a parameterization with a single set, are convenient. Location and scale are ambiguous in statistics and have intuitive meanings.
The distribution can be written in more than $n$ parameters. There is no need to express the distribution which requires $n$ parameters in exactly $n$ parameters.
The set of location parameters, requiring $n-1$ parameters, can be expressed as an Euclidian coordinate on the unit sphere, in which case there are $n$ instead of $n-1$ parameters, of which one is superfluous parameter due to the constraint. You could leave out one of the coordinates or express it in terms of spherical coordinates.
Why not instead define the distribution for one unconstrained parameter i.e. $y := \kappa \mu \in \mathbb{R}^p$?
There's nothing that stops you from using $\vec{y} = \kappa \vec{x}$ as an alternative parameterization. You can do it if you like, but it is not done because it is not useful.
Expressing the position in terms of an Euclidian coordinate makes you add an extra parameter to the parameterization. This is a redundancy and you could use the redundancy to mix the $\kappa$ into the location parameter.
But, that is not why the redundancy has been created. The reason to use $n$ location parameters instead of $n-1$ is because the expression $\vec{x} \cdot \vec{\mu}$ is easier than the trigonometric functions when we would use spherical coordinates.
The constraint on $\mu$ does not relate to $\kappa$. It is not necessary to get the $\kappa$ mixed into the redundant parameter.
The $\mu$ and $\kappa$ have a physical and intuitive meaning. But, mixing them together into one single parameter does not seem to make sense.
There are various types of parameterizations for distributions and various reasons to use them. In your comments, you have mentioned four.
(1) notational succinctness
The notation of the distribution function $\frac{1}{2\pi I_0 (\kappa)} e^{\kappa \vec{x} \cdot\vec{\mu} }$ is very succinct when we express the location as $\vec{x}$.
Succinctness is exactly the very reason why we use the notation of the location, which only needs $n-1$ parameters, in terms of $n$ parameters with a constraint. Using spherical coordinates, or expressing just the first $n-1$ coordinates of the $n$-dimensional vector $\vec{x}$ would reduce the parameters, but make the formulas more complicated.
Also, notational succinctness is not a goal in itself, but it should be a tool to reach another goal. In this answer there are some ways to reparameterize a 3 parameter distribution family in terms of 2 parameters. Yes, we can reduce the number of parameters, we can make parameterization more succinct, but it has no use in itself.
(2) avoiding confusion (e.g. my own)
There are indeed many distributions that have confusing parameterizations.
For instance, the geometric distribution expresses the probability of the number of Bernoulli trials until a successful trial is observed. The number of trials can be counted including the successful trial, or counting only the unsuccessful trials. This leads to the use of two different supports.
For instance, the non-central parameter in non-central distributions can be expressed in different ways. It can relate to a shift of the mean of the normal distributions from which the distribution is derived, or it can relate to a shift of the mean of the distribution itself.
For instance, the scale parameter in the normal distribution. Often we use the notation $\mathcal{N}(\mu, \sigma^2)$, which parameterizes the normal distribution by the mean and the variance. But some computer codes (for instance R) uses instead of $\sigma^2$ as the parameter, the standard deviation $\sigma$ as a parameter.
In the case of the von Mises distribution I find the parameterization in terms of a separate location parameter $\vec{x}$ (with the constraint that it is on the unit-sphere) and a scale parameter $\kappa$ not at all confusing. Combining them together would be more confusing.
(3) reducing computation
Many distributions have parameters that are easier for computations.
I do not see how a computational simplification would be possible for the von Mises-Fisher distribution by using $\vec{y} = \kappa \vec{\mu}$. You state that you encounter an updating relation
$$\kappa_{nk} \mu_{nk} = \kappa_{n-1,k} \mu_{n-1,k} + \pi_{nk} o_n$$
Indeed with the alternative parameterization, you would get
$$\vec{y}_{nk} = \vec{y}_{n-1,k} + \pi_{nk} o_n$$
How did you get this parametrization? For your computations you could, of course, always switch to an alternative parametrization. The conversion between the single parameter $\vec{y}$ and the two parameters $(\vec{\mu},\kappa)$ is not so difficult.
I might also suggest that $\kappa \mu$ has a physically intuitive meaning
Physical meaning is another reason for the choice of parameterizations.
Like in the previous example with the beta distribution, the $\alpha$ and $\beta$ parameters are are useful for computations, but the mean and variance can be interpreted more easily.
In distributions like the exponential distribution and the Poisson distribution the 'rate' parameter is used in place of a 'scale' parameter.
The 'rate' has a direct meaning relating to the physical process. The rate parameter is a good choice to describe the properties of the physical process that creates the distribution.
The rate is the inverse of the 'scale' parameter. The scale parameter is often a preferable choice to describe distributions. It relates to the variation and to transformations of data.
You mention that in the case of the von Mises(-Fisher) distribution the combination of the $\mu\kappa$ parameter as some physical meaning like the combination of location and intensity, with the example of pulling a table cloth.
What this example lacks is a direct use or application of this physical meaning. For example, the peak height of the normal distribution is $f_{max} = \frac{1}{\sigma\sqrt{2\pi}}$. This has a physical meaning, namely the peak height, but should it be used just because it has a physical meaning?
