If we're going to talk about common terminology, then it seems wrong to ignore the fact that the majority of UI guidance talks about "click" and "alt-click" far more than they talk about numbered mouse buttons. This has been true since macs went to 1 button, since laptops with 3 buttons became very hard to find, since mice with 5-12+ (yes, really) buttons because available in every mainstream computer store, and since laptops with 2 buttons became virtually impossible to find -- to name just a few major inflection points. At this point, the most common interface device has 0.5 buttons -- the "tap/click/etc" action of trackpads. Mice (and their related desktop bretheren) have bifurcated between 2 buttons plus a sometimes-clickable scroll wheel (ubiquitous in lower-cost setups) and devices with many physical "buttons" (typically, 2 "main" buttons, a clickable wheel, and at least 2 other buttons). If this doesn't match your intuition, I invite you to do an internet search for "computer mouse" and then for "gaming mouse".
This change to "click" is not an accident: naming the concept after the action rather than the tool has been recommended practice from UI/CHI researchers since at least the early 90's. UIs generally can't count on 3+ buttons, and most users don't internalize that many simultaneous modes anyway, so "click" and "alt-click" are defacto standards. Even now, the "alt click" is losing its hold, since touch-based interfaces are the baseline for many users' experince, and alt-tap is cumbersome for most of those.
To put this in more concrete terms: the last 3 devices I've used as my daily driver, covering 15+ years and a solid 5-digit hours of computer use have been incapable of directly producing a "middle mouse click", even with modifier-chording tricks that belie the term "emacs pinky". The same is true of nearly all of my peers and coworkers over that period, including programmers, prose writers, and "creatives".
~Chad