Renaming the Greek "Middle" Voice?

I agree with Carl Conrad that the term “Middle Voice” creates the false impression that the real contrast in Greek is between active and passive and that the “middle” voice is something of a misfit in an otherwise clear system. I have been thinking about what term could be used to replace “middle” that would avoid this implication and better fit the actual usage of what we have all been calling the “middle” voice.

Here’s my problem: The Greek middle voice is clearly NOT equivalent to the English reflexive construction, but it IS very much like the reflexive of some other IndoEuropean languages. I happen to be a fluent speaker of Spanish, so examples from that language are very easy for me to produce, but the same is true for French and Italian according to what I have read on those languages.

Here are a few Spanish reflexive constructions with English equivalents. Notice that there is a good deal of difference between the two languages in their use of reflexives. All of the Spanish sentences have reflexive constructions. Many of the English equivalents do not.

Me corté.
I cut myself.
Me corté el dedo.
I cut my finger.
Me voy.
I’m leaving.
Me compré un nuevo reloj.
I bought myself a new watch.
¿Te diste cuenta que Alfredo ya llegó?
Did you realize that Alfredo has arrived?
Se despertó el bebe.
The baby woke up.

Of course I could write hundreds of these examples easily, but I think this is enough to make the point. Reflexive constructions vary widely between languages. The Greek “middle” voice is very much like the reflexive of Romance languages, but quite unlike the English reflexive. If I were writing a Greek grammar in Spanish, there’s no question of what I would call the “middle” voice: la voz reflexiva. But calling it the “Reflexive Voice” in English could cause serious confusion since many “middle” voice Greek verbs require active voice English translations, not reflexive ones.

So, what should we call the “middle” voice to avoid the confusion caused by the term “middle” and also avoid the confusion that could be created by calling this voice “reflexive” in English?