She didn't invent the term gentle parenting. It was well in use on forums by the time her 2016 book was published. Plus there was an earlier book (2005) with the title "Adventures in Gentle Discipline" which was well-known on every Attachment Parenting forum because it was in the La Leche League approved library. I found one earlier than this (mid or late 90s IIRC) using the term, not in the title though, but it doesn't really resemble what most people today would call gentle discipline/parenting, plus I forgot the title of this book, so can't reference it.
I wrote an article about this which was never published, so I really did look into this in some detail - I looked up on google trends, plus the google thing which searches books, MN advanced search and the advanced search of every parenting forum I could find which is still live, as well as the wayback machine and the timeline seems to be that the word "gentle" starts being used in conjunction with either discipline or parenting sporadically throughout the 2000s, then a bit more activity from 2010-2012, 2012 is the start of a slow but steady incline which peaks very suddenly in 2021 which I think is probably when influencers got in on it.
MNers started using it in about 2012, though argued about what it is a lot and mostly agreed it's a silly name, and it was definitely a thing on UK parenting sites long before it became used as a term on more American-centric sites like Reddit (though this might also be that Reddit had very little parenting content until much more recently, and a lot of the other places people discuss/ed parenting are either gone or were behind logins - ie deep web, e.g. LiveJournal, or email newsgroups). Before 2012, there were still threads with the exact same concepts, it just got referred to by either specific book titles e.g. How To Talk, or Taking Children Seriously or Unconditional Parenting, or descriptions like "the thing where you don't use punishments" which is what I definitely called it a few times! Also bear in mind that even today, in some parts of America smacking/spanking is still apparently a very normal and accepted thing - so they tend to have a bit of a different understanding. I have seen some people on more US based sites refer to Supernanny or 123 Magic as gentle parenting, whereas neither are IMO.
SOS started the "gentle" line with the gentle sleep book in 2015, which makes sense because there was a gap in the market for a UK sleep alternative - lots of big popular sleep trainer types had books out, the main alternative at the time was Elizabeth Pantley, and the idea of being "gentle" with sleep made sense. She then used that as a sort of brand (or, wikipedia suggests that her publisher did). Nothing wrong with doing that, and it's clearly made her books very recognisable. But, again, she didn't invent the term and it may have even been accidental - or possibly she was hoping to ride the wave of the significance of the word being used to denote the sort of "follow on" from attachment parenting which it tended to be at that time, which if that was the aim, it was very successful.