We have to distinguish between labels and recognized attributes. In the case of being a Franciscan (or in my case Benedictine), there is a formal process of attachment to what is a recognized charism, and Order, in the Church. A signed oblation charter submitted to the abbot, along with a promise made in front of the community made me an Oblate. No one can claim to be an Oblate without going through the process.
By the same token, a layman is a state of being in the Church. The Church has a formal definition of what a layperson is, or rather isn’t (not ordained and not professed religious).
The problem with the “traditionalist” label (or “liberal”, or “modernist”) or whatever is that it is usually one of self identification or worse, being applied to another to essentially say “thank God I am not like them”. Yes partisan spirit is at the root. But alas, the label is the weapon. Labeling stigmatizes folks, particularly like in the case of “traditionalist” where there is no clear Church-approved definition. Proof of that is that it goes beyond just a preference for the EF Mass. Some extend it to rejecting all or part of Vatican II, to outright dissent and disobedience (SSPX), to schism (sedevacantists, etc.). Summorum Pontificum doesn’t use the term “traditionalist” to identify those preferring the EF Mass.
We really should refrain from any label not formally recognized by the Church. Human nature is such that labeling, no matter how well-intended, can end up being misused to exclude others who test our comfort zones. This is not just in religious circles.