Hi @Sylvain_Juge.
I have the same behaviour for elastic apm agent. Name for first transaction is ok, other transactions have broken name.
I tried open-telemetry agent. I have no problem to start Payara with this agent, as you mentioned before. Distributing tracing is not working for 100%, I still have a issue with asynchronous requests... but this is another topic. 
I have a question about elastic java agent and this issue which is in this thread. Should I raise some ticket for Elastic Team or what is your opinion how to resolve this bug? Or Could you share this bug with your team members?
Thanks.