"Core" is "Bitcoin". NOBODY else in the entire cryptosphere calls it "core"! The entire world agrees upon this other than those who decided to invest in the altcoin bcash and potentially its spinoff, now they're protecting it with religious zealot fervor, even after weathering 80%+ losses.
As many rational minds have stated, if Bcash wanted the least bit of respect, they would change their name to something other with "Bitcoin" in it, and stop trying to win support through confusion and non-stop attacks on bitcoin.
You obviously move in other parts of the cryptosphere to me, I see neutral people referring to 'Bitcoin core' all the time, to distinguish it from 'Bitcoin cash'. Neutral people don't take sides, and the history of the split from my point of view is it's far too early to know what the eventual outcome will be, but btc and bch can both use the name 'bitcoin' because they share the same genesis block. Which version is closer to the original and therefore has the best claim to Bitcoin name is an opinion, not a fact, so for now both should use it, but add meaningful word after Bitcoin to help distinguish them. Only zealots on one side or other would disagree, probably to stifle honest discussion.
What would really help distinguish Bcash is if they removed the word "bitcoin" from their name. That sure would clear up a great deal of confusion. BTW, "neutral" people don't refer to it as "Bitcoin Core" either. It's pretty obvious that if BCH was indeed "the real bitcoin," it would have proven itself as such by now. Instead its just managed to sink in price in terms of BTC, and now its going to further fracture its user base. I'm quite honestly not invested in either one, I just think Bcash's approach to marketing itself has been extremely dishonest.
You are correct, my Grandma just says Bitcoin (no core or cash after)