For me I'm happy just leaving this to the player to decide as I said above. I'll call for the Charisma check (if the declared action meets the requirement for calling for checks) and the player applies whichever skill proficiency he or she thinks is appropriate.Interesting position.
To me, it's not about skill at lying vs. truth. It's telling a convincing lie or targetted omission of the truth to get what you want vs. using the right arguments/rhetoric/appeals to the heart to convince someone to get your way.
I've known quite a few people who suck at lying but are very persuasive, so that part falls flat to me.
To me, the skill check is Deception here because the player and character know that they are using a true statement in a deceptive manner. Can they sell the deception? Did the character control their body language, their eyes, their poker face, etc.
Even if the player is picking Persuasion over Deception solely because the bonus is bigger, I just can't be given to care. As @cmad1977 mentioned, it's unlikely the variance will be so great that it will make a huge difference once we factor in the swinginess of a d20. When taking that plus a general desire to move things along into consideration, I don't see any value in trying to parse whether a true statement used as a way to deceive should have its uncertainty as to outcome resolved via Deception or Persuasion. In short, I'd prefer to just get on with it than spend time on this at the table.
Assuming we have good faith on the part of the player, this approach to adjudication eliminates 2 of the 3 issues the OP raised and others no doubt that may arise in play, particularly around disagreement on which skill or tool proficiency applies to the ability check.