It seems that 'schooner' has multiple meanings depending where on the globe you are, but apparently for the U.K. refers to the larger of two glasses (the smaller being a 'clipper') which are used for consuming sherry. Comes from maritime connections, apparently, and refers to the size of ships which plied between Spain and the U.K., importing sherry. When sherry drinking went through a bit of an upsurge in the '60's, these 'schooners' were a popular way of drinking irresponsibly (so nothing new then) - and I think these boobs were designed as a means to prevent the glass from slipping through your hand, should you have been less than sober. Nothing better than holding onto boobs if you're in trouble - at least they can cook, do the washing up, and change a fuse in the plug.

Only joking ladies.
And now for the serious part..........as far as I know, all glasses which additionally have stems and feet (i.e. not tumblers), are made by making the bowl first, so I guess if you cock up the bowl part you'd bin it and wouldn't go on to make the stem and foot. We're talking hand made pieces only.
Otis - in your original note you made no use of the word 'pontil' - although we are assuming there is a ground/polished pontil on the underside of the base. Neither the pontil depression or the rings of which you speak would appear to be a specific design feature - but I'm sure are simply part of the construction process when this glass was made.
Your..........."rims around the base of the bowl can be detected"...........is possibly that optical effect seen within the glass when the bowl is viewed at a sort of oblique angle. I don't know what these rings are called, but they are common on much modern clear glass - seen easily on round clear decanters if you look down the side of the piece. I suspect simply an optical effect caused by the shape (the roundness) of the piece, and can be seen in your glass which is clear and round.
Best of luck in finding the maker.