Otherwise why would I pay for support if I could just self-host? They can strictly define the parts they are willing to support, though.
He doesn't update his version of your software and this leads to a security issue.
Are you liable?
I feel people really try to pick the worst possible interpretation of these laws just so they can hate on the EU.
What is software in this case? The open source software you developed as such or only the Installations you sold support for?
This part isn't clear enough to confirm you assumption.
It could mean the software instance installed by your support customer, it could mean the software as such you intend to sell support for, no matter if you are really get paid for support or not.
The word you were looking for is "support".
If there is something wrong with the "supported" open-source software, then you may expect a certain level of "support". Full-stop.
That generally entails an SLA that says your issue will be reacted to within N-time of opening the issue, which might be nuanced by the tier-level of support purchased. That you are provided access to documentation, or even the source code itself. You might be provided with best-effort support by an agent, which is limited to resolving documented defects, or configuration, or acknowledging standing-bugs which cannot be resolved.
What you cannot expect is the software is updated in accordance with the support incident. For that, send patches, or pay somebody to send patches.