Now I have verified the behaviour.
If there is no IP address in VLAN 1, the syslogging won't fully work.
When I enter an IP address in VLAN 1, the syslogging works (in VLAN 235). Note that the VLAN 1 does not need to be up or have to have any active ports. Just having the IP address configured enables the logging.
As the VLAN 1 does not have any active ports in my configuration, that VLAN does not even appear in the routing table. This is correct by definition.
Once the logging is working, removing the IP address from VLAN 1 disables the syslogging after 2 messages.
When entering the VLAN 1 IP address, the switch sends some (10-20 maybe) *old* messages to the syslog server, and starts logging of new messages. Today, for example, when I entered the IP address, the switch sent syslog messages from last Friday to the server.
I was wrong when I said that the source address was the old IP address of the VLAN 1. This is not true. The source address is the address of the VLAN interface where the syslog server is located (this is how it should be). I was mislead by the fact the the 5300XL switch includes the IP address of VLAN 1 in the syslog message, for some reason (and the switch logs two messages even after removing the IP address from VLAN 1).
So, to conclude:
- 5300XL and E.07.22
- VLAN 1 has to have an IP address for syslogging to work
- No active ports in VLAN 1 are needed
- Management VLAN is not needed
- Primary VLAN setting does not affect
- Logging server can be in any VLAN
- This feature is not listed in the Management and Configuration Guide (for E.07.2x). Please correct me if this can be implied from some chapter.