[net/refactor] Rework ThreadOpenConnections logic

Summary:

Make the connection counts explicit and extract into interface functions
around m_conn_type. Using explicit counting and switch statements where
possible should help prevent counting bugs in the future.

Partial backport (16/19) of core PR19316:
https://github.com/bitcoin/bitcoin/pull/19316/commits/7f7b83deb2427599c129f4ff581d4d045461e459

Depends on D8725.

Test Plan:

ninja all check-all

Reviewers: #bitcoin_abc, majcosta

Reviewed By: #bitcoin_abc, majcosta

Subscribers: majcosta

Differential Revision: https://reviews.bitcoinabc.org/D8726