There's no trick allowing a creature proficiency with armor
This reader is unaware of a handler being able to use the skill Handle Animal to teach a creature a trick that grants the creature actual proficiency with armor (but see below). For most mounts, this DM urges masterwork studded leather (PH 123, 126) (for a Medium humanoid 175 gp; 20 lbs.; adjust accordingly for bigger and littler nonhumanoids)—which has no armor check penalty—and not worrying about it.
However, you may want more than that.
The Dragon #292 (Feb. 2002) Wizards Workshop column "Sage Advice" includes the following exchange:
When you purchase barding for a mount, is it proficient in the armor, or does the mount suffer the armor check penalty?
Anyone wearing armor suffers the appropriate armor check penalty. Trained war mounts are proficient in light, medium, and heavy armor, [sic] untrained mounts are not proficient in any kind of armor, and they suffer additional penalties for untrained armor use as noted on page 80 of the Player's Handbook. (110)
(The column is unsigned.) So far as this reader is aware, this information is never repeated anywhere in the entirety of the official Dungeons and Dragons, Third Edition corpus.
Keep in mind that trained war mount (also war-trained—q.v. the Ride skill use Fight with Warhorse) is itself a slippery and ill-defined term in the core rules. It takes the Rules of the Game Web column "All about Mounts (Part Two)" (Feb. 2005) to explain that the purpose combat riding (described in the skill Handle Animal) makes a creature into a trained war mount (or war-trained or trained for battle or whatever), although the column doesn't use that language exactly. (Also note that those columns are dismissed by some fans as unreliable, sometimes in part but sometimes as a whole.)
But neither animals wearing armor nor mounts wearing barding is addressed in the columns "Animals" (Parts 1–4) and "All about Mounts" (Parts 1–5) (these latter, in some cases, being the only source of some information about mounts). The core rules, as the question notes, aren't much help either. Indeed, it is a pretty miserable oversight that the core rules aren't clearer on this when the game has as one of its central classes a warrior astride a mighty steed.
To bring this all together, this DM recommends that the purpose combat riding gives the creature proficiency with light, medium, and heavy armors (yet probably not the associated feats—there are already enough uses for the power psychic reformation) and that part of the price of the warhorses and the war pony includes them being trained for the purpose combat riding.
The armor trick exists, but it doesn't do what it probably should
The Masters of the Wild (Feb. 2002) armor trick says, "The animal is willing to accept the burden of armor" (18), but it later explains that
Horses, ponies, riding lizards, and riding dogs typically accept armor in the form of barding, but wild creatures simply refuse the burden. With the armor trick… a character can adapt any animal to the use of armor. (36)
Yay? So the armor trick is not a way for the animal to gain armor proficiency but a trick tax on the masters of unusual creatures. This DM recommends ignoring the armor trick completely: masters of unusual creatures typically have enough training issues already.