What is fair is to be open with your cofounder and investors.

Should I just keep the vested equity?

Should the other stockholders just dilute your holdings?

Is a cash buyout common or appropriate?

Probably less common that telling you to pound sand. Cash is the life blood of a company and giving you cash is very damaging to the startup.

To put it another way, you need to negotiate. To negotiate you need to care about the interests of the other people and in the case of a startup recognize that the vast majority of the work is in the future. And that there is no value in sunk costs. Good luck.

Why not offer your stake to the company’s investors at the previous valuation?

The general thinking is that the new team is essentially a new business. This new business needs cash, and taking that away is more or less a non-starter at this point. Fundamentally, founders are compensated for seeing things through, not for partial work. It's a really hard sell to either ask a cash-poor company for cash for non-productive purposes or to ask new investors to put up cash that's going towards an exit-package (ie. someone who's not going to be contributing). The only leverage you may have is how much pain you can cause by not being bought out, which puts you in an adversarial relationship with everyone else. I don't think that's good for you.

So I'd recommend you'd just hold on to your vested shares to build good will. Indicate your intent to exit, but you like the new idea and wouldn't put additional pressure on the business at this stage. If the business is successful in the future, you may get bought out by later stage investors to simplify the cap table and reduce governance complexity. But for that outcome, you'd need to be on good terms with current investors and the new team.