If someone modified the database and deployed it publicly (or as part of a webapp that was deployed publicly), they'd need to share the modified source, yes. AGPL is roughly GPL but where the traditional definition of "shipping software" is expanded to include deploying on a network.

My guess is that they have two motivations, both of which are fairly traditional GPL motivations: 1) sell AGPL exceptions to commercial licensees; and 2) prevent a competitor from making a private commercial fork, where the competitor improve the DB and licenses their version to clients, without sharing the source to their improvements.

To be pedantic: deployed it a an external service, not necessarily a public service (I suppose to a legal entity that would not normally share license rights, such as an inividual not part of the organisation or to another organisation). Those external users would have to be given an option to access the full source with modifications.

As for the "part of a web service"-bit, I'm not sure what the agpl's actual "reach" is. My understanding is that with a (modified) db under agpl powering eg a web app runing in php, the end users (accessing only the web server) would not be entitled to the db source. If the agpl covered the web srrver itself or a php library on the other hand, the users would be entitled to that code?

Similarly if one sold a modified db-as-a-service, modifications would be covered by the agpl.

While its interesting to theorize about databases and AGPL, I am rather sure there hasn't been any actually case or a legal reason to think that the database would become a derivative work when used together with a webserver.

If it was, the EULA for SQL server and oracle would have to include copyright permission for derivative works. That we do not see that should be a clear sign that the scope of copyright has not reached that far yet.