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.
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.