Update liveness probe to support authentication token

What does this MR do?

The Liveness Probe was updated in !404 (merged). The changes made came with the following issues:

• No support for authentication token
• Can return multiple occurrences of name in config.toml

This MR fixes both issues and add integration tests for each of them

Why was this MR needed?

Fix an issue introduced in v0.58.0

What's the best way to test this MR?

Integration tests must pass

What are the relevant issue numbers?

fixes gitlab-org/gitlab-runner#36959 (closed) #507 (closed)