These screwdrivers were called ‘Contour Grip’ by Snap On, but the name doesn’t seem to have been used much since introduction, so most folks just seem to refer to them as the ‘classic’ or ‘square handle’ models.
Think they were introduced in the early 60’s. They replaced the earlier ‘Octo Grip’ drivers with the distinctive 8 sided handle. The Countour Grips were made of ‘Tenite’, an Eastman material that’s pretty decent. There has been some adverse publicity concerning Tenite degrading, but in my experience it was only the odd bad batch.
These were certainly offered until the late 80’s. They were certainly ‘industry standard’ when I was working at a flying club in 1986. I bought a few in about 1987 and they are superb. Wish I’d bought more!
Standard handle colours were black (for slotted and Philllips tips) and red (for Pozidriv). Small batches were obviously made in other colours (I’ve seen yellow) but I was never offered these new. I’ve also seen handles of I believe the same material, but with a triangular shape. I think these were short lived, and seldom seen in the U.K.
I’ve bought one or two used versions of these drivers, and the material is so durable that (provided there are no cracks) you can sand and re polish the handles to re finish them!
This pattern stayed in production till about ‘88, when the “Power Grip” was introduced. (That’s the design that we think of as the “hard handle”). There may have been tweaks in the design at some point, which might be why the site the OP found suggested an end date of 1970, but it’s all a bit before my time!