I've used the hot nichrome wire technique to cut thin glass tubes neatly, for instance from fluoro light tubes. Works fine.
As others mentioned, scribe a fine neat line around the tube, then wrap a taught nichrome wire once around the tube right on the scribe line. Apply current to wire. If you got it right there will be a faint 'tink' after a few moments, and a clean cut. No need for any water.
A good way to hold the wire, is to make a kind of springy bow, with insulated ends and the wire stung across them. With enough slack that you can loop it around the tube with the wire tensioned by the bow. It HAS to make even contact with the glass all round, so the wire better be free of kinks. You can make it nice and straight by hand drawing (stretching) the wire out a bit.
To scribe the line around a tube, take a piece of tough paper and wrap it tightly around the tube. When it's straight the end edge will line up with itself. Tape it in place and use the paper edge as a guide. Anything that can score glass will work. Fine sharp-edged file for instance. Or diamond tip glass cutter. The scribe line is the most critical part - it has to be even and straight all the way round. Roller wheel glass cutters won't work on a glass cylinder by hand - you can't apply the force evenly enough. Might work with some kind of roller jig.
That picture of the failed crystal... I'm wondering how the solder joins worked. The crystal disk has gold plating on both sides, and usually there are some kind of clips that contact the plating, one per side. In that one, were the mounts soldered to the flat faces of the gold plating on each edge?
Maybe it just failed due to fatigue from the oscillation warping? Perhaps over driven?
In any case, I doubt you can solder back to the gold plating, if it didn't already come right off the crystal.
Also it was probably in a vacuum.
Glass enclosed crystals are pretty. The one below is (I think) from 1966. 5.0000MHz