edit: @ Delioth & Talvieno: You can drain a volcano, you just have to be aware that it will periodically rise back towards whatever level it was at embark, you could seal it off entirely at the desired height, or keep it low with a pump or 2. I have vaguely heard rumours that sealing it off can cause it to push the magma onto your floor anyway, dont know nowadays. The extra magma doesnt exactly 'glitch' in. It's travelling via a kind of flow/pressure mechanism working much like a river. In rivers water doesnt actually move. New water entering the upstream map edge teleports to the downstream edge where the water is less than 7/7 and then trickles off the map, the river body never actually moves. A volcano refilling is rather like this, with the source being (i think) the magma flow tiles directly above smr, it spawns new magma and puts it in the first available spot, the surface of the magma pool in the caldera.
Perhaps this was true in 31.25 or earlier? When I was experimenting with a volcano, I tried seeing if I could empty the whole thing above my traps into the caverns. It didn't work. The magma above a certain point seemed to be suspended on an invisible floor, while the rest of it poured nicely and neatly into the caverns as you would expect it to. The upper layers of the magma in the volcano remained entirely unmoved.
Going back with a savescum (I was just experimenting, don't hate me) and emptying it higher up successfully drained the upper layers of the volcano, but with an odd effect. Every few ticks a new 1/1 magma tile would appear suspended at the very top of the volcano, and fall down to where the magma was - it was basically raining magma from the lip of the volcano. That was something I really didn't expect at all, and I'm not sure how it makes sense. It's possible my map was just severely glitched, but I'm not sure what would have caused it.
Finally, in a river, water usually
does move (but only at the end) because water enters the map more slowly than it drains. After a time, your water level will start to go down towards the exiting end, and you'll see the water start to actually flow. On a small map with a stream, you could eventually see the water level go all the way down to 5 or 6 across 2/3 or more of the stream. A better way to observe this behavior is in arena mode. Just turn it on and wait for a while - despite there being a very large number of "in" tiles (nine), and only three "out" tiles, you'll see the water level at the top of the waterfall quickly start to drain.