Gerrit has already a parameter to define an identifier for the instances. This could be included in the events generated by the different nodes.
The instanceName is currently set in the gerrit.config. It defaults to the full hostname if not set in gerrit.instanceName.
Propagation of the id in the events can be done as explained in Proposed solution or Alternative Solution - 2.
instanceName is not used to uniquely identify an instance in a cluster.
For example, it is, available to the email templating system.
Using it would couple the presentation logic to something more backend specific as identifying an instance in a cluster. Also, it could not necessarily be relevant for the end users receiving it in their inbox.
Same consideration as in the proposed solution