refactor(db): use semantic pooler probe identifiers

[seonghobae]

Scope

Canonical bounded naming repair for the private PgBouncer/PgCat admin-console probe on protected main@8dc746920c12988f082e914879d95e13c9693535.

The verified defect was an organization-owned private nested helper _run() plus generic connection/query locals (dsn, password, conn, cur, row) inside the database-pooler detection boundary. The surrounding product context already supplies unambiguous bounded vocabulary: pooler probe, pooler admin console, and version row.

TDD lineage

• RED 9efcdbf28476be4f0f81162b37ea93bb952567d4: AST contract requires _run_pooler_probe and semantic pooler connection/query identifiers while rejecting the old generic names. Exact protected base has _run, dsn, password, conn, cur, and row, so the regression corresponds to the current defect.
• Repair be55ea51c32fc73990ddf3870b797ee5bf029e0a: _run → _run_pooler_probe, dsn → pooler_dsn, password → pooler_password, conn → pooler_connection, cur → pooler_cursor, row → version_row, with all private callers updated in the same change.

Compatibility / persistence

No public FastAPI dependency, SQLAlchemy engine/session symbol, configuration key, PostgreSQL schema/table/column/index/constraint, Alembic migration, DSN wire format, pooler protocol command, read/write routing behavior, timeout, locking, or persistence behavior changes. SHOW VERSION; and the public get_pooler_detection() contract are unchanged.

No database migration, FK/index rebuild, ORM remap, UPSERT change, hot-partition change, rollback transform, or read/write-topology change is required for this private-identifier repair.

Verification boundary

Fresh exact-head backend/security/review evidence is required. Historical base or predecessor evidence does not transfer. Keep Draft until the unchanged final head has all live required checks terminal-success, valid findings/threads resolved, qualifying independent approval where required, and ordinary protected-branch governance permits merge.

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