Yes I understand what your saying.
But even given that I see the term “tertiary” far surpassing even the use or not of the name “third order”. (The “third” had usually to do not with “levels” but rather with the order of foundation historically --though this was not always the case. Though for that matter the first “tertiaries” can be said to have been of the humiliati) (the Vatican Congregation that over sees them all refers to them in general as “Third Orders Secular” which has both terms contained in it)
I myself would
prefer the term and do --* “secular order”* or similar – to “third order”. I find the emphasis on the secular character of the vocation to be brought out more fully for any of the Orders
But having said that I still – surpassing and even in a way beyond the term “third order” – I find the term “tertiary” to be a venerable term that can still be used (and is) similar to other “general vocational terms” terms such as “religious” “knight”“nun”“hermit”
A “secular” does not work well in that more general fashion or that general vocation term.
Hence I find the term
tertiary to be splendid and beautiful. Like
hermit.
And make use of it in my writing and conversation (you of course never have not use the term

)
In doing this though --I do it
while still calling of course– Secular Carmelites –
Secular Carmelites and Secular Franciscans —
Secular Franciscans and Secular Augustinians –
Secular Augustinians and so on and on…