extensionSearchUsed should be incremented when the extension $search is executed and legacySearchUsed should be incrememented upon legacy $search execution. These should not be incremented on the parsing or query shape paths, only during execution.

Pattern match to the respective $vectorSearch testing:

• Extension-enabled/disabled paths
• non-search/$vectorSearch does not affect these counters
• Explain increments counters once per attempt
• Sharded per-node counter semantics