Document output saving public APIs

[ChrisRackauckas-Claude]

Ignore until reviewed by @ChrisRackauckas.

Summary

• Add missing rendered docs entries for SavedValues, IndependentlyLinearizedSolution, and LinearizingSavingCallback on the output-saving docs page.
• Expand source docstrings for the same output-saving storage/linearization APIs with signatures, arguments, returns, and examples.

Audit

• Export/public declaration search found exports only; no public, Expr(:public), or @public declarations.
• Mechanical audit reports no exported names missing doc bindings.
• Mechanical docs-page audit reports no exported names missing rendered @docs entries.

Local verification

• timeout 3600 julia --project=. -m Runic --inplace src/saving.jl src/independentlylinearizedutils.jl
• timeout 3600 julia --project=. -m Runic --check .
• timeout 3600 julia --project=. -e 'using Pkg; Pkg.test()' passed.\n- timeout 3600 julia --project=docs docs/make.jl timed out in Documenter.HTMLWriter.render after document checks and HTML rendering had started.\n- A targeted output-saving Documenter build with edit links disabled also timed out in Documenter.HTMLWriter.render after document checks.\n