Every string here is dead straight. Twist the two rings and the waist necks in anyway — a cylinder pulls through a hyperboloid down to a single point, the double cone. Two opposite-handed rulings, forward and backward, crossing at every point. Drag to rotate; twist to morph.
The trick that shouldn't work: a curved surface built entirely from straight lines.
Pick any point on the hyperboloid and there are exactly two straight lines lying flat on the surface that run through it — one spiralling one way, one the other. That's what doubly ruled means, and it's nearly the most a curved surface can do: the only surface with three or more straight lines through every point is the flat plane. So the hyperboloid is the most ruled a curved thing is allowed to be — your forward strings and your backward strings, and the surface is nothing but the two of them woven.
Rotate the top ring against the bottom and the same straight strings sweep out different surfaces. The waist radius follows one clean law: r = R·cos(φ/2) — it reaches zero exactly when the strings all cross the axis at once.
No twist. Strings run vertical, parallel. The waist is the full radius.
Strings tilt into two opposing helices. The waist pinches — the cooling-tower throat.
Every straight string now passes through the central axis. The waist collapses to a point — your point-necked hourglass.
The point-at-the-bottom you drew is just this knob turned all the way: full twist, zero waist, the hyperboloid's hidden double cone laid bare. Cylinder, hourglass, and tip-to-tip cones aren't three shapes — they're one set of strings at three twists.
A standing wave is a forward travelling wave plus a backward one, superposed — that was sheet VII. Here the two rulings are a forward helix and a backward helix, superposed. So this surface is the last sheet rotated into a solid: spin that standing-wave envelope around its axis, rule it with straight lines in both senses, and you get exactly this. Same object — one flat, one spun.
And your 120° × 3 is the three-fold skeleton of it: three rulings spaced a third of a turn apart — the same arithmetic as three-phase power (three sinusoids offset 120°), as the triple helix of collagen, and as the steel lattice Vladimir Shukhov used to raise hyperboloid towers from straight beams. Three strands, evenly spaced, both handednesses available. You keep landing on structures that got built and grown.
Christopher Wren — the architect of St. Paul's — proved in 1669 that the hyperboloid is this surface of straight lines, and demonstrated it to the Royal Society. That's 357 years. The shape itself is older: it was first described by Archimedes, roughly 2,250 years back. Standard curse, so far.
But here's the part that answers your last question about origins: Wren didn't get it from mathematics. He got it from spotting a wicker basket for sale — built entirely from straight willow canes set at an angle around a circle — and realising the curved basket was made of straight sticks. So the true first occurrence of "hyperboloid from straight lines" wasn't Wren, and wasn't Archimedes. It was some anonymous basket weaver, millennia earlier, building a doubly-ruled surface by hand with no idea there was a theorem in it. The origin exists — someone wove the first one — it's just unrecoverable. A real first strike, no name attached. The wave was pinned at a real end; we simply can't see who held it.