I have always thought that, when dealing with multinomial logistic regression, the idea was to linearly model the "logistic" functions of the probability densities of the different response categories (as explained here). I put quotation marks around "logistic" since we do not use the real logistic functions, $\log\left(\frac{\pi_j(x)}{1-\pi_j(x)}\right)$, but with denominator equal to the density of a certain "pivot" category.
I then discovered the extension to a log-linear model, where the logarithms of the probability distributions, $\log(\pi_j(x))$, are directly modeled, as explained here. At the end of the day, it is still very similar to the previous assumption.
However, I had problems when I discovered the existence of this alternative formulation, which is explained in the R
package mlogit
's vignette (pdf). Basically, every $\log(\pi_j(x))$ is modeled with $\alpha_j+\bar{\beta}\cdot\bar{x}$, where $\alpha_j$ is an intercept which is characteristic of every response category, and the vector of linear coefficients, $\bar{\beta}$, is the same for every category.
My question is not maybe a real question, but why this ambiguity?