This is also my understanding of why the AGPL was created to try to close the loophole of running GPL code on the backend; since running code on a server doesn't entail distributing the server binary to people who access the service, the GPL didn't enforce any requirements on sharing any modifications to code that was modified and used as part of the backend for the service. Some felt that this went against the spirit of the GPL, so a new license that made this also explicitly disallowed was created.

As for the LGPL, my recollection is that it differs from the GPL in creating an exception for dynamically linking to the compiled binary; the phrasing of the (non-L) GPL requires that any "derivative works" of the GPL-licensed product (I can't remember if that's the exact term, but it's a something similarly general) must abide by the GPL, which could be interpreted as applying to something that dynamically links to a library licensed under the GPL. Licensing something under the LGPL gives permission for users to dynamically link to your product without requiring them to share the code linking to it.