when an agent asks to see your next project

I take that to mean query the normal way, but mention that the agent wanted to see it right at the start of the e-mail.

That's how I've responded to that sort of request, anyway. I guess I figured if they meant a full ms, better to have myself corrected on that than the other way around.

I take it to mean query, but start by mentioning the last ms and the request to see more of your work. That's how I've handled it.

What they said.
The agent thought your writing showed promise, but didn't think the first manuscript was saleable.

Agreed with all the above comments.

When I ask to see a next project it's generally because I think the writer has talent but the first project wasn't quite ready or wasn't something I thought would sell well.

I say send it. She made the statement before closing to queries so you got the go ahead to do it regardless.

Either send it with a cover letter explaining that she asked to see the next project, or first send a sort of hybrid query, starting with the explanation that the agent had asked to see the next work, and it's done now, but you understand that the agent is closed to submissions, and is the agent still interested? You could include a query or an even briefer pitch after that.

I did something like that with an agent, who very graciously asked me to send the manuscript along in circumstances where she wouldn't have been interested in looking at it otherwise. It didn't end with an offer of representation, but she gave me some good feedback and encouraged me to submit my next next project to her.

If she asked to see it, send it.