So the responses in this thread have been very helpful. Now that I'm home and can lay hands on the freewheels, here's what I'm seeing.

A) Two Regina freewheels with gold-colored cogs, both hand-marked "FF", both have two groove, a BSA bottom bracket cup starts to bind after about a turn and a half on either one.

B) One Regina freewheel with silver cogs, not marked "FF" in any way, has a single groove, a BSA bottom bracket cup threads onto it smoothly until it bottoms out.

C) The SunTour Winner 7-speed freewheel from post 6 above, hand-marked "FF" on both a cog and the back of the freewheel mechanism, a BSA bottom bracket cup threads onto it smoothly.

So, as I indicated above, I think the hand-markings on the SunTour freewheel are just wrong. I had previously not noticed any marking on the top of the mechanism, but with a little digital assistance here it is:



Does the "GE" there (at the 10 o'clock position) mean anything? What about the squiggle below that?