Precisely that.
A more relevant example might be something like my web-facing software, `bitcoin-top`, which I release under the AGPL.
https://bitcoin.esotericnonsense.com
You are free to copy the source code and run it on your website, as a part of your website (e.g. link to it on a subdomain), etc.
In that case the matter of whether you have to offer the source is kind of immaterial - it's the same source I'm offering and have mirrored at least 4 times, and in this year 2019 we don't need to be shipping CDROMs about.
I don't believe that your statement:
* If you don't modify the AGPL program, you don't have to offer source.
is strictly correct, rather that it's simply the case that if you don't modify it, you're already offering source, you can just point people to my github or whatever.
If you modify it and serve a different version, say a litecoin node monitor or you tweak and improve it or whatever, you must release the source code because our software would be licenced under the AGPL.