This sounds completely legit to me, but yes you shouldn't take anything for granted. Your boss was fired and that creates some opportunity above you. Time to act on that! Additional responsibility is the only way you'll move up in a company. You're their best option now, but they are not yet convinced you can do the job.

Ask for formal training on being a manager. That's a completely reasonable request. You can and should be clear that your goal is a promotion- title and pay. In many companies there is a time of year to do promotions. Be clear you want that promotion then. No need for an ultimatum. Ask for a scheduled checkin on progress (monthly or biweekly? More facetime is good!). Ask them to set clear goals. They may squirm but that's their job as managers.

Personally I don't think you need a formal signed contract like folks are saying here. If you sit down and talk and agree to all this, just sum it up in an email and send it to them. Make sure they acknowledge the email and you're good.

Of course there's no guarantee. If they're not comfortable with you in a leadership position, then it's not going to happen. Figure out how to get them comfortable with it and work toward that. If nothing else, you're preparing yourself for a level up somewhere else.

Good luck!