Problem: Argument type mismatches are excluded from EmmyLua checks.

Solution: Correct runtime annotations and calls, document local checker
workarounds, and enable the diagnostic.

AI-assisted

Enable EmmyLua's assign-type-mismatch diagnostic after correcting the
runtime annotations and assignments it flags.

Correct option, tuple, and return types, and use separate locals when a
value changes representation. Add guards and narrow casts where needed
so the runtime passes the stricter check. Regenerate the affected docs.

AI-assisted

Problem: LuaLS struggles with the generics used in Nvim's runtime,
requiring broad diagnostic suppressions. Indexing is also slow.

Solution: Use EmmyLua for type checks in the build and CI. It offers
more sophisticated type checking, substantially better support for
generics, and much better flow analysis.

Correct the affected annotations. Use `@internal`, supported directly by
EmmyLua, instead of `@nodoc` for shared internal declarations, and
support it in the help parser.

AI-assisted

docs: update instructions for debugging LSP

Previously, it was suggested to set:

    vim.lsp.log.set_format_func(vim.inspect)

This made sense before f72c13341afb3e2459202d3f6e57cd00614c2ab5, when
`format_func` was called once per argument being logged, but since that
commit it's called with the log level followed by the other args, so the
suggested setting would call `vim.inspect(log_level, ....)` which would
just print the human readable name of the current log level and no other
details, for example with this set I saw in my logs:

    "DEBUG""DEBUG""DEBUG""DEBUG"

Instead just rely on the default formatter, which will:

> ... log the level, date, source and line number of the
caller, followed by the arguments.

Close #38431
Close #38521
Close #38530

Co-authored-by: tayheau <thopsore@pasteur.fr>
Co-authored-by: zeertzjq <zeertzjq@outlook.com>
Co-authored-by: Olivia Kinnear <git@superatomic.dev>