I don't agree with that definition. Mainly because the game that "started" the Metroidvania bandwagon plays nothing like the Castlevanias before it most of the time, mirroring more-or-less closely only one of them: Simon's Quest. It's a term created to describe a collection of gameplay mechanics and elements, not a "genre of Castlevania".
For me, this is what the term "Metroidvania" describes:
-You can level up, meaning that your character gets stronger or more resistant;
-Related to the point above: Focus on combat, where the reward is getting stronger.
-You must acquire key relics to progress on the game;
-Equipment is awarded to the player, but is not mandatory to reach the end;
-Related to the point above: Said equipment may be mixed and matched to deal with different situations. "Stronger" doesn't always means "better", and this is also not required, but optional;
-Inventory management is part of the progression, and usage of potions or temporary buffs is encouraged.
-You have a map, and this map shows you how much of the game you have covered;
-Non-linear progression is present;
And then, I disagree with you that this term should be used to describe games like Axiom Verge, because I think Axiom Verge is a Metroid-like, not a "metroidvania". It lacks the high customizability of equipment combination, the inventory management, the leveling, and the focus on combat. Its focus on exploration and puzzle-solving is much more prevalent through the game.