Apparently there are lots of regional names for power tools.
Talked to my carpenter. He needs one of these. The scrolling knob on top allows more precise cuts.
One puts a fiendish death trap on your head that you have 10 seconds to escape from, because you didn’t always feed your pet gerbil his favorite chow.
The other just chops off your head and is done with it.
While we’re at it, in the UK (should you find yourself east of the pond and in need of powered cutting tools), a jigsaw means the one that sits atop a sheet of wood and is hand-guided through potentially intricate cuts, and - because Milwaukee aren’t the biggest of names here (though they certainly have a presence) - Sawzall isn’t used so much: the coarser, more demolition-oriented ones are known as reciprocating saws.
Jigsaw has been used to mean bench-mounted reciprocating saws into which wood is fed (and still is used that way, given the right context), but if you talk about a tool as a jigsaw, people will expect something hand-held.
This is my experience too.
To sum it up, it can’t be summed up. 
There is (even just from the small sample “people who have posted in this thread”) frequent fundamental mismatch in the way these tools are named. In one area, “X” means a particular kind of saw, while in another area, the same “X” can never mean that kind, because it means something else.
Conclusion: I declare that no one may ask me for any type of electrically-powered saw without showing me a reasonably accurate picture of it. 
Table saw, band saw, jig saw, sawsall, radial arm saw, coping saw, rip saw, crosscut saw; I’ve got them all.
That’s all, continue what you were doing.
What, no compound sliding miter saw? Amateur!
I’m so embarrassed. Forgot that one and the chain saw as well.
What about a see saw?
Nope, but does a c-clamp count? Keyhole, Japanese saw, dovetail saw, hacksaw, plus the power saws mentioned above by Fir na tine and others. Seems like a good dick measuring competition to me. Next up: blade tools!
OK, then, zip saw. They were all the rage in the late 90s, and then I think the company making them must have spent everything on their marketing campaign and disappeared. They looked like drill bits, except they cut sideways instead of penetrating forward.
The Sawzall isn’t a craftsman’s tool; it’s a tool for apes to foul things up quickly. 