No worries and no judgement. I just happen to know how to interpret and decipher the colophon or copyright page.

The name on the spine says Fawcett because that’s the imprint or trade name; you can think of it like a brand. Ballantine bought out Fawcett and chose to keep the Fawcett and Gold Medal Books brands as an imprint for a while (probably because they had a large catalog by that point). Ballantine is the publisher, Fawcett is the imprint, and Random House is the parent company (and all of it is owned by Penguin now).

It isn’t “based on” the first printing or, at least, that’s only partially correct. Editions usually get updated with new formatting, fonts, cover art, commentary, and possibly light editorial revisions for typos or printing mistakes (or in this case introducing printing errors). It’s the same intellectual work, it’s just been rereleased. I don’t think there are any other pre-Ballantine Fawcett editions because they were acquired pretty soon after this particular title was published. In some of my browsing for this title I saw that it was also published by the New English Library company in 1981 in the UK.

The first edition of this from Fawcett is out there, but it’s oddly expensive for a paperback: https://www.lwcurrey.com/pages/books/169068/robert-a-heinlein/the-number-of-the-beast

The hardcover is even more pricey: https://www.alcuinbooks.com/pages/books/024016/robert-a-heinlein/the-number-of-the-beast

The edition information usually doesn’t matter unless you’re a bookseller or a cataloger or a nerd. One thing is usually true: first editions usually don’t say they are first editions; they just have a copyright date.