Let's start from the beginning. Once upon a time, someone decided it'd be neat if you got a little extra bonus for rotating pieces into tight spots. In The New Tetris (1999), twists (or "spin moves," as it called them) would trigger an avalanche effect. The method it used to detect if a twist was performed is pretty simple: if the piece cannot be moved left, right, or up from its final position, it was twisted into a tight place. This method of twist detection is typically called the "Immobile" algorithm. (Diagram, an example of an Immobile twist)
A little bit later down the line, "spin moves" get retooled into t-spins for Tetris Worlds (2001). The t-spin -- as the name implies -- can only be performed with the T piece. Instead of triggering a cascade, t-spins are rewarded with a score bonus. The algorithm for detecting t-spins is quite a bit different than the simple Immobile approach. Other than the requirement that the piece be a T, there are two other criteria. One is that the last move -- other than forcing the piece to lock into place -- must be a rotation. The other is that three of the four cells diagonally adjacent to the center of the piece must be occupied. (Diagram, cells in question marked in red) As these cells are the corners of the bounding box the piece rotates in, this algorithm is typically referred to as the "3-Corner" method of t-spin detection.
One of the consequences of 3-Corner and the wall kicks featured in SRS -- the guideline's standard rotation system -- is that it is quite possible to have a move that can be reached by a straight drop be recognized as a t-spin. This diagram demonstrates the so-called "EZ T-Spin" that confused many Tetris DS players who accidentally triggered it.
Of course, it seems a bit silly to reward a move that didn't require a twist with a bonus reserved for twists. The Tetris Company's first answer to this problem was to disqualify moves that used wall kicks from the t-spin bonus. This eliminates the EZ T-Spin well enough, but it also took a few other twists with it. This neat little twist requires a wall kick in SRS, so it lost its t-spin status. The controversial t-spin triple is obviously unreachable without a kick as well, so it too went unrecognized. The Imperial Cross was still viable as the final position is not reached directly by a wall kick, but other setups were not so lucky.
Losing so many interesting moves on account of one problem case was unsatisfactory. The Tetris Company's next course of action was to introduce the T-Spin Mini, which earns less points than a "proper" t-spin. This is where things start to get really confusing.
For a while, my understanding of t-spin minis was that they were t-spins that cleared one or no lines and were reached by a wall kick. This is how everyone tends to explain them, and it seems to be consistent with the kick/no kick problem in general. I could have sworn this was their explicit definition in at least one game, but now I am not so sure. Whatever the case may be, they are pretty darn weird now.
Determining the definition of a t-spin mini was like a puzzle in itself. I went into my tests holding onto the tenet that the wall kick is what made a t-spin mini... mini. I came out realizing this -- although central to the original problem -- is almost completely irrelevant. In fact, -- bewilderingly enough -- a wall kick is one of the few things that can make a t-spin not mini.
My first draft of t-spin mini rules was that the t-spin be kicked the the final (x,y) position, one or no lines be cleared, and -- based on some one particular kick t-spin single that wasn't mini and some experimentation with rotating from "proper" t-spins to other states -- that the piece end in a position other than point-down. I provided this set of diagrams as evidence. Unfortunately, I found that things stopped checking out as I delved deeper. Upon testing a kick t-spin single using the t-spin triple twist, I found that it no longer demoted the move to mini status. I hypothesized that it had stricter requirements for kick t-spins to be counted as "proper," like a 4-corner test or an Immobile requirement. Neither of which checked out. To make matters worse, some of my experimentation led me to discover that wall kicking wasn't related to a spin being demoted to mini status at all.
Tetris Concept member Zaphod offered up his draft of the potential rules, citing that a spin was mini if one of the open corners on the 3-corner check was on the side with the point of the T. However, this case seemed to act as a counter-example. He also suggested that lines cleared did not come into consideration as a t-spin double would assuredly have the two point corners filled. I countered this premise with an esoteric t-spin double setup. Zaphod ultimately proved that orientation was irrelevant with a case like this. We also ultimately concluded that there Immobile was not used in any way in exempting minis, and that there was a test specifically for the t-spin triple kick.
To restate our conclusion, a t-spin is mini if there is an open corner on the point-end, it clears one line or less, or reached its final (x,y) position by way of the t-spin triple wall kick. If you count the the three conditions for a t-spin to be checked in the first place, there are six rules to evaluate a twist. It could be worse, but these rules are already severe overkill in trying to nullify EZ T-Spins. All they needed to do to detect that was add an Immobile check.
In fact, you kind of have to wonder what the point of 3-corner is in the first place. Do we really need to disqualify this and other perfectly good twists? In fact, with the advent of scoring bonuses for twists with other pieces in games such as Amagami Tetris and Tetris Party Deluxe, it probably makes a whole lot more sense to go back to simple Immobile than try to adapt 3-corner to pieces it doesn't really fit. Rather than pile on more exceptions and introduce more eccentricities, I think it is about time The Tetris Company switched back to the algorithm they had at the start.
(Parting note: These tests were done on TetrisFriends.Com, which means these rules should be fairly up-to-date with the guideline. Since we just figured this out recently, I have no idea if previous games with t-spin minis follow the same rules or match the commonly held belief that kick = mini. Similarly, I couldn't tell you if more recent games with spin minis have revised these criteria.)