Update.php asking for clean SQL DB!

I reinstalled 6.14 because the system status report said I had a severe security breach potential. After running status again, I get:

Database updates Out of date
Some modules have database schema updates to install. You should run the database update script immediately.

However when I run the update script, it askes for DB password, etc, and then says it can only install into an empty DB. I glanced at the URL and it's running install.pdp.

What's going on here? Is the first error message misleading? The site seems to be function OK.