A lot of them, for my taste way too many.
basic walking and stops, amagues, corridas, the cruzado system, ochos, ochos cortados, ganchos, sandwichito, giros, step variations and basic walking of Milongas, and loads of other things I even forgot the name. I can do all of this somehow, but I would prefer to have learned just 3 of them but right.
Example: The week after we learned ochos cortados we did only Milonga walking techniques, never coming back to ochos cortados. I that week, in technique courses (same school), we were practicing basic Tango walking. To me a mixture of 3 complexe things just leading to a chaos in my head.
An alternative perspective (very tango nuevo

): The whole tango vocabulary is three things: step, pivot, switching from cross to parallel system and back. Everything else can (admittedly with some contortions) be assembled from these three elements.
Vocabulary is not something to be learned, but it is worked out examples of how these basic elements can be combined. Amagues is walk+stop, just with tricky timing, corridas are walking. ochos are pivot+walk (or, in the way it is usually taught cross-system, pivot, walk). giro is pivot+walk, ocho cortado is giro+amague, gancho is an adornment for walk+stop, and so on, and so on. So i think to some extent they are not there to be learned correctly, but to practice pivot, walk, switching systems in different combinations and variations, and your teacher might feel that presenting different variations helps people being flexible in how they combine pieces. Like in a boxing gym, where one day you work jab-cross-hook, and another cross-hook-cross, then jab-jab-cross and so on - less working combos, more doing drills.
Sometimes i wish navieras analysis of tango would have filtered down to teaching - i wonder if students would have an easier time if e.g. giro, calesita, ocho would be introduced as being the same move (find the followers axis, pivot till they point the desired direction, take them off the axis in one of the 3 possible directions, repeat). Having names for everything does reify things.
Re: What vocabulary: I think a useful progression would be:
1) walk + stop on single axis, same track
2) rockstep ( this includes stopping on the center axis, some idea of rebound, amague)
4) using the invisible weight shift on axis to move between systems (walk and rockstep in all systems, playing with doing this kind of double time move everywhere to see how switching systems works on all three tracks)
5) calesita (find and establish followers axis, and how to get out of this), ocho (chained small calesitas), giro (unfolded ocho). (plus playing with rocksteps and walking and rebounds in those - for practicing precision in positioning i like "long ochos" - a ocho with three steps instead of one between pivots)
I don't really think the cross is a good move in the beginning - it is finicky, there are a few different ways to lead it, and more a practiced piece of choreography. It is a very useful piece of choreography, but if we want to look at tango from a nuevo / first principles perspective it is somewhat of an outlier.