fix(docs): select repository-aware Pages deployment settings

[Christopher Anderson (christopheranderson)]

Description

Port the final reviewed deployment fix from microsoft/project-rayfin#2063. Exact public-repository CI selects the https://microsoft.github.io/rayfin/ target, while local, fork, and private CI retain their existing deployment settings. Adds cache inputs, local opt-in documentation, and regression checks for generated assets, canonical URLs, navigation, repository links, and search paths.

Validation:

• cd docs/site && rushx test — passed (11 tests).
• rush build --to-except docs-site — passed (13 operations).
• cd docs/site && DOCUSAURUS_PUBLIC_SITE=true rushx build && DOCUSAURUS_PUBLIC_SITE=true rushx test:build — passed (5 artifact tests).

Types of Changes

Surface Area	Public	Experimental	Internal
CLI	[ ]	[ ]	[ ]
Copilot plugin	[ ]	[ ]	[ ]
SDK	[ ]	[ ]	[ ]
Templates	[ ]	[ ]	[ ]
Docs	[x]	[ ]	[ ]
Container	[ ]	[ ]	[ ]
Host	[ ]	[ ]	[ ]

• Spec only
• Other: Docs-site deployment configuration and regression tests.

Hotfix Needed

• Hotfix needed
• Release tags applied

Checklist

Public changes

• TypeDocs updated
• Guide docs updated
• AI files updated

Experimental changes

• SDK changes: exported from experimental exports path only
• CLI changes: gated behind feature flag
• Documented only in experimental docs folder

Internal changes

• TypeDocs marked as internal
• SDK changes: exported from _internal exports path only
• CLI: no public surface changes

Related Issues

Port of microsoft/project-rayfin#2063.