feat: revamp server logging

[mhuisi]

This PR revamps the server logging mechanism to allow filtering the log output by LSP method.

[leanprover-community-bot]

Mathlib CI status (docs):

• ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the nightly-with-mathlib branch. Try git rebase 80b8e44072aa9f940c74b71b57386ce22fc9ea2b --onto 14ff08db6f651775ead432d367b6b083878bb0f9. You can force Mathlib CI using the force-mathlib-ci label. (2025-10-15 13:28:44)
• ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the nightly-with-mathlib branch. Try git rebase dec007693a6a0e097623158a95642cc78cdff97f --onto efbbb0b230ce95653d25b59c83fd24a51a8bf363. You can force Mathlib CI using the force-mathlib-ci label. (2025-10-24 15:07:26)
• ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the nightly-with-mathlib branch. Try git rebase dec007693a6a0e097623158a95642cc78cdff97f --onto 3a42ee0c3060e2a8365927d205ead5990366053d. You can force Mathlib CI using the force-mathlib-ci label. (2025-10-28 11:38:37)

[leanprover-bot]

Reference manual CI status:

• ❗ Reference manual CI will not be attempted unless your PR branches off the nightly-with-manual branch. Try git rebase 80b8e44072aa9f940c74b71b57386ce22fc9ea2b --onto d3dda9f6d4428a906c096067ecb75e432afc4615. You can force reference manual CI using the force-manual-ci label. (2025-10-15 13:28:46)
• ❗ Reference manual CI will not be attempted unless your PR branches off the nightly-with-manual branch. Try git rebase dec007693a6a0e097623158a95642cc78cdff97f --onto d3dda9f6d4428a906c096067ecb75e432afc4615. You can force reference manual CI using the force-manual-ci label. (2025-10-24 15:07:28)