There might be a A quota/rate-limiting policy on service 2 endpoint which limits calls to 10 per second or within service 2, there might be a thread pool or worker pool that handles only 10 concurrent requests. check for Any SpikeArrest, Quota, or ConcurrentRateLimit policies and also check Service 2’s configuration: max threads, max connections, or concurrent requests etc. (Service 2 might have a resource bottleneck aswell) .

One point to be noted is that if response time is more than 30s, apigee X (55s in apigee edge) by default treats it as a timeout (maybe 11th request takes a larger window due to resource timeouts in backend)

The Apigee documentation you shared is about rate limits imposed by Apigee on its own management APIs—not on your proxy/API traffic.

ie; calls made by Apigee UI, CLI or REST API (Authenticated) to Apigee’s own APIs, such as Creating or updating API proxies, Deploying revisions etc
the 6000 per min limit means You can make up to 6,000 management API requests per minute per project (organization).If you exceed this limit, you'll get 429 Too Many Requests errors.

this does not explain your 10-successful-calls + 1-timeout pattern , that issue is likely due to concurrency limits, connection pooling, or backend rate limits .