[FEATURE]: Revisit aligning `chatcmpl-<ID>` with other notions of request ID throughout

[rmccorm4]

Feature request

With ongoing tracing work, we will likely have notions of trace_id and/or request_id flowing around. On the client-side when talking to OpenAI APIs, they will get back things like chatcmpl-<RandomID> in the responses. We should align that ID in client side response with internal request/trace ID logs for easier tracking, if it makes sense.

Relates to #2243

CC @nnshah1

Describe the problem you're encountering

n/a

Describe alternatives you've tried

No response