Actually, it's currently calculated exactly the same as a spell attack: 1d20 + charisma + proficiency... which is exactly the same as a charisma check. I'll change the wording.
The crazy thing about binders is that just binding a different set of vestiges already makes you very different. I'm keeping some notes in the margin on this one, which I'll be providing to the DM who runs the game this will be used in, and one of the key notes is that a binder needs to discover the name, formal title, and seal of a vestige, so a DM can limit access to vestiges by requiring the players to go on a quest to find those details, which are likely to be lost or closely-guarded knowledge (especially for the higher-level vestiges). This helps to get binders diverging a little, and then there's the choice of Trusted Vestiges, which helps too. In 3.5, there was also the Pact Augmentation effects you could choose from every few levels, which allowed you to tailor the character even more, and I've just added that as well.
Incidentally, I was initially considering doing a more general Summoner class, with subclasses Binder (binds vestiges into items, making things like possessed armour or sentient swords), Manifester (gives vestiges physical form in a manner similar to Summon Monster, giving you minions for the day) and Channeler (channels the powers and skills of the vestiges through themselves, like the 3.5e Binder) but I decided that was too complex when all I needed for now was a way to convert our party's Binder character into 5e. I'll probably still create that Summoner class though, just not yet. Stay tuned!
Yeah, I don't want it to steal the spotlight, but I do want it to be able to get a little bit of the flavour of the various classes. It's partly because of the lore - when you're binding the soul of a former rogue, you should feel a little bit roguish, and when you're binding the remnants of a mad demilich, you should get a taste of what that feels like, too. Also, this ability to be a jack-of-all-trades, dipping a toe into all the classes without ever fully being any one of them, is a sizeable part of why our Binder chose to play as a Binder, so if I remove too much of that then it defeats the reason I'm building this class.
I've updated the first post with my latest additions; I'm currently trying to come up with a decent 20th-level capstone, and starting to convert the vestiges from Tome of Magic. There's a lot, but I'd like to get the first two or three levels done soon, because then we can start playing (our Binder is level 5).
As always, critique and suggestions welcome.