Is a model fitted to data or is data fitted to a model? Is there a conceptual or procedural difference between fitting a model to data and fitting data to model? An example of the first wording can be seen in https://courses.washington.edu/matlab1/ModelFitting.html, and of the second in https://reference.wolfram.com/applications/eda/FittingDataToLinearModelsByLeast-SquaresTechniques.html. 
 A: Pretty much every source or person I've ever interacted with except the Wolfram source you linked refers to the process as fitting a model to data.  This makes sense, since the model is the dynamic object and the data is static (a.k.a. fixed and constant).
To put a point on it, I like Larry Wasserman's approach to this.  In his telling, a statistical model is a collection of distributions.  For example, the collection of all normal distributions:
$$ \{ \text{Normal}(\mu, \sigma) : \mu, \sigma \in R, \sigma > 0 \} $$
or the set of all Poisson distributions:
$$ \{ \text{Poisson}(\lambda) : \lambda \in R, \lambda > 0 \} $$
Fitting a distribution to data is any algorithm that combines a statistical model with a set of data (the data is fixed), and chooses exactly one of the distributions from the model as the one that "best" reflects the data.
The model is the thing that changes (sort of): we are collapsing it from an entire collection of possibilities into a single best choice.  The data is just the data; nothing happens to it at all.
A: Usually, we assume our data corresponds to the "real world" and making any modifications means we are moving away from modelling the "real world". For example, one needs to take care removing outliers since even if it makes computation nicer, outliers were still part of our data. 
When testing a model or estimating properties of an estimator using bootstrap or other resampling techniques, we may simulate new data using an estimated model and our original data. This makes the assumption that the model is correct, and we are not modifying our original data.  
A: In the field of Rasch modelling it is common to fit the data to the model. The model is assumed to be correct and it is the analyst's job to find data which conform to it. The Wikipedia article on Rasch contains more details about the how and the why.
But I agree with others that in general in statistics we fit the model to the data because we can change the model but it is felt to be bad form to select or modify the data.
A: Typically, the observed data are fixed while the model is mutable (e.g. because parameters are estimated), so it is the model that is made to fit the data, not the other way around. (Usually people mean this case when they say either expression.)
When people say they fit data to a model I find myself trying to figure out what the heck did they do to the data?.
[Now if you're transforming data, that would arguably be 'fitting data to a model', but people almost never say that for this case.]
A: I think 'fitting the model' sounds wrong to some because the model structure does not change as part of the fitting process. But the model parameters do, which is why it is correct.
Fitting is still an awkward term and it vaguely sounds like some underhand manipulation is being done. I much prefer to just talk about parameter estimation.
A: This could also be seen from Bayesian perspective and goodness of fit. We might interpret "model fitted to data" as in finding out probability of parameters fits the given data well i.e, $p(\theta|X)$ a posterior and  "data fitted to a model" as in finding out probability of observing the data given the model i.e.,  $p(X|\theta)$ likelihood.
A: Interesting topic here and I could not let it pass me by without saying a few words. Here is my take on this. From a scientific point of view, a model by definition is a conceptual representation (universally accepted as a reference) of an event or process. With that in mind, it is my understanding that any data out there is a dynamic entity, meaning it changes over time. Unlike statistical models that are static(reference) and were developed to describe certain phenomena. Having said that, the data at hand is also a representation of an event but that is yet to be validated due to its dynamic nature. The process by which the data is validated (compared with a model/reference already in existence) is what we call data modeling or model fitting of the data. In conclusion, we should rather be saying "FITTING the DATA to A MODEL."
