They are largely related, but they are not the same thing because the pieces involved behave differently in different situations. (The technique is named after the piece so you have "z spin", "s spin", and "t spin".)
I went to look up z-spins but the google results suggested you must have meant t-spins.
The "z spin" in the earliest incarnation was actually a coding error allowing only the "z" shaped piece to spin "through" a one block hole if the piece could have otherwise fit in the lines beneath that hole. (It required extremely precise timing.) The error became legend which became a commonly implemented extension in "Tetris" clones and official "Tetris" games. I think, but I'm not positive, that every official "Tetris" game has offered it intentionally in at least some modes.
Yep. That's a "t spin" alright.
Watching a video of one t-spin just looked downright wrong to me.
When the timing is "loose" it is largely just a "funsies" thing. The vast majority of "Tetris" clones that implement the technique (intentionally) make it unavailable in certain modes. One of the best puzzle based versions, a Java game if I recall, forced players to spin through fixed (blocks that can't be destroyed) layers to clear off every line with one piece.)
When the timing is "tight" it becomes experience based making it a useful feature in challenge or competitive modes. This is, of course, mostly limited to intentional implementations because lines cleared with complicated spinning get you a lot more points. (The points awarded for spinning multiple pieces into place over consecutive lines clears is usually worth more than a tetris clear.)
Yet the vast majority of "Tetris" clones do and have done for over a decade.
I would certainly not allow that.
This is a lot like the other technique called "spinning". (It has been called other things; I call it this because it is what I've always called it.) The other "spinning" technique is where a lot of "Tetris" clones allow a piece to spin forever (always moveable never becoming fixed) as a way to counteract the essentially instantaneous drop at higher speeds. That "spinning" technique offered a way to solve the original games problem (At higher speeds it was impossible to move the piece to the edges of the play field.) without unnecessarily limited the fall speed. When this "spinning" technique is poorly implemented it decreases the difficulty of the game to absurdly low levels making it really not a fun a game to play.
That is a good website.
most of the proper rules for tetris titles are listed here
It should be noted that the website suggests, as most "Tetris" clones do in any event, allowing, recognizing, and rewarding players for "t spin" moves.
So, whether you allow it or not, some variation of the "? spin" moves are extremely common and if going to be supported should be considered from the start.