The Crimson Binome
Hero
All proficiency checks are a sub-type of ability check, where you add your proficiency if it's relevant. There's no such thing just a Lute check; it's either a Dexterity check or a Charisma check, and you add your proficiency bonus if you're proficient.I don't believe tool proficiencies generally benefit from ability score bonuses, whereas a skill does.
As far as I can tell, the Performance skill completely supersedes all instrument proficiencies, if you have it. If you are proficient in Performance, then you are effectively proficient in every single instrument, since you will always add your proficiency bonus when using an instrument to perform. It's just that, since individual instruments also count as tools, it's possible for someone to learn how to play one without otherwise being proficient in Performance.