If I may offer a counter-argument, as a fan of the genre: If we're absolutely going to stick with the term "Roguelike", we should be using it for games like Rogue, the PC game from 1980. We shouldn't just through it around willy-nilly for any game with permadeath and randomly-generated levels. A roguelike, ideally, should look something like this:
Generally speaking, a roguelike should present the player with a top-down (or at least heavily visible) view of the game world – invariably a randomly-generated "dungeon" – that features on-screen enemies and items scattered across the floor. The game
must, except in the absolute most extreme edge-cases, be turn-based, and should let the player accumulate an inventory of randomly-generated items – some of which are not immediately identified – which can affect their stats and other entities in the game world. Every time you play the game, the world, items, enemies, and level geography
must be randomized.
It should be... you know... like Rogue. A dungeon-crawling, turn-based adventure with a heavy emphasis on random generation. A game that focuses just as much on careful preparation and item accrual as it does on the frequent necessity for immediate improvisation.
Games like The Binding of Isaac, Hades (1 or 2, take your pick), Diablo, and other action-heavy games absolutely should not be lumped in with what I've described above, because they're not the same at all. They're completely different in terms of gameplay. "Roguelite" is an alright descriptor for them, but when you call those games "roguelikes", it's like me calling a a basketball game a platformer because you run and jump and collect items in it. It's vaguely similar, but it's an extremely poor, inaccurate descriptor.
Action games, even if they have some element of randomness to them, aren't roguelikes, because they aren't like Rogue. I personally dislike the term "roguelike" because it's applied far, far too generally – I think "Dungeon Crawler" is a much better name for the genre, and associates it closely with one of its most immediately-recognizable and -understandable properties (Mystery Dungeon).