I do agree that the terminology, as so often is loose and not something to get hung up about.
But I'll add one thing to add to the discussion, which is to think about how to hold these saws. With the sort of small saw that Steve has just made, it's normal to have a long handle at one side and a smaller knob on the other. It's natural to hold the saw with the long handle in your dominant hand and the knob in your other, a bit like you would hold a large file. But it's not always the best way. As ever, we have variables such as the type of wood being sawn, its thickness and how it's orientated. So if your first grip doesn't feel right, try a different one.
Watching the video of Frank Klausz (who is clearly a very experienced craftsman worth watching closely) he starts off with the larger saw holding the frame above the obvious handle, which gives him better control to keep the blade vertical.
When he swaps to the other saw, he uses a two handed grip, with one hand wrapped around the other, enclosing the handle.
I suggest it's worth experimenting with both those grips, on any size of framed saw - the best one is whatever works best for you.
And here are a couple more illustrations of the two handed grip.
This is from Charles Hayward's "Tools for Woodwork" from 1946
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And these are from Simon Barley's 2016 "British Saws" - I think it's from an old tool catalogue - where I think it's safe to assume that "Old Clegg" would have been quite well-informed about saw handles:
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