Fully qualify index names

What does this MR do?

This change comes with a small refactoring to extract a Index class and re-arranges things a bit. This is mostly in preparation for selecting the index to be rebuilt automatically.

The functional change is that we require index names to be fully qualified, i.e. you must specify the schema name. This is mostly to remove an assumption about public being the schema from the codebase (we typically don't assume public).

Note it's fine to assume public in specs.

Issue: #250324 (closed)

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content