"goobledegook = e + 5.6. Wow you didn't know about goobledegook? Let me teach you about this revolutionary concept." I majored in mathematics and can say I've never heard of versine and have probably used it many times without knowing. I don't think everything needs a name.
I think the point of these was for when they read low-precision values out of a printed table and then multiplied them using slide rules. Even though the article doesn't say it, I can't think of any other practical reason to multiply by adding logarithms.
The log tables let you work without slide rules, which were themselves horribly expensive not too terribly long ago in the grand scheme of things. But yes, this was all about practical calculation, not mathematics for its own sake, that needed to be carried out quickly by people who wouldn't have handy access to machines that keep the dirty arithmetic details (and potential for error) hidden from the operator, whether that's your ship's navigator or the machinist with nothing but a sine bar and gauge blocks setting up a precision operation, and if you needed precision using a four-figure table book, the only way you were going to get it was if the derived operations were in the tables.