refactor(meta): reorganize code of global barrier manager (part 2)

[wenym1]

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What's changed and what's your intention?

In #19334, we refactor the code of global barrier manager by simply moving the code to different files, without modifying any of the logics.

In this PR, we further refactor the code of global barrier manager to hide the visibility of some structs or fields to have better encapsulation.

Checklist

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