I followed the below URL to setup event provider on Adobe I/O for AEM site events.
When i tried to create the second integration, i can't find the event provider of my AEM site to show up in experience cloud
Request you to please provide some more details on the step where you encountered the problem and what was it?
Refer this article: Experience League Intelligent Guidance | Adobe
Could you provide more details when you say "second integration" ?
Do you mean to say that you can't see "Adobe Services" group?