I think Yales' terminology is appropriate, though I find it may be more prudent to refer to the program being attempted (undergraduate, graduate, master's, doctoral, post-doctoral, etc) and just append First-Year; is it important to dictate which year they are in aside from the first?
If more specificity is truly needed, Last(Final?)-Year may also be applied in that way, or 'Last-Year Graduate Student.' If in the middle years, state the year as suggested. That seems more than appropriate.
First/Last go together, as opposed to Initial/Final, so First/Last would be at least consistent in its lexicon.
Such as a First-Year Undergraduate being a literal first of the first year.
A First-Year Graduate student being the first year of the graduate program.
My own criticism of my suggestion is that it may not translate into other academic systems cleanly if their progression structure is different. I can only speak for the American way.
I cannot yet comment so I apologize for responding to the other answer and if inappropriate can be edited out by whomever. In regards to 'themself', it does give me a bit of trouble in syntax. Despite being in both communities it is relevant (linguistics and LGBT), it feels wrong to say out loud. In my own writing and having to reference, speak with and about gender-neutral individuals, 'themselves' is read just as well and can be the same as 'themself' depending on the syntax of the sentence. "They completed the task themselves." as long as the article is defined already, it sounds appropriate. If it isn't, it will sound, again, a bit strange. "Ash completed the task alone." resolves the issue completely, and avoids repetitive pronoun usage.
This is a failure of English with the lack of a third person singular neuter. Many other languages have it. Apologies again for the digression, but it may still be valuable in gender-neutral academic writing. I would appreciate any comments or perpsective on this.