The momentum wavefunction (or more properly, the wavefunction in the momentum basis) completely determines the position wavefunction (wavefunction in the position basis). And we can probe the momentum wavefunction with any particle at all, by setting up identical (say) electrons and seeing the momentum they impart on a variety of test particles. That is to say, the probability distribution of momentum of a particle does not depend on what we use to probe it.

As the position wavefunction is now completely determined in a probe-agnostic matter, it would be hard to justify calling a probe that didn't yield the corresponding probability distribution a “position measurement”.