proxy: Report warm cold start if connection is from the local cache

[khanova]

Problem

• quotes in serialized string
• no status if connection is from local cache

Summary of changes

• remove quotes
• report warm if connection if from local cache

Checklist before requesting a review

• I have performed a self-review of my code.
• If it is a core feature, I have added thorough tests.
• Do we need to implement analytics? if so did you add the relevant metrics to the dashboard?
• If this PR requires public announcement, mark it with /release-notes label and add several sentences in this section.

Checklist before merging

• Do not forget to reformat commit message to not include the above checklist

[github-actions]

2520 tests run: 2396 passed, 0 failed, 124 skipped (full report)

---

Flaky tests (1)

Postgres 15

• test_vm_bit_clear_on_heap_lock: release

Code coverage* (full report)

• functions: 28.7% (7028 of 24495 functions)
• lines: 47.4% (43411 of 91504 lines)

* collected from Rust tests only

---

_{The comment gets automatically updated with the latest test results
6329ca5 at 2024-03-13T11:25:43.563Z :recycle:}