docs: prepare v0.3.0 docs for release

[eagle-head]

Summary

Makes the documentation accurate and self-consistent for cutting v0.3.0, so the release can be tagged with no further doc edits. Found by a read-only doc-readiness audit and verified against the source (and git) before fixing — versions/deps were already correct (0.3.0 in both packages, engines.node >=22, peer react only), so this PR is documentation only.

Changelogs (core + react + docs site)

• Record the breaking minimum-Node bump (>=18 → >=22 on root/core; the react package newly declares engines.node >=22).
• Record the single-emit-per-transition behavior change (each transition now emits exactly one snapshot).
• Record that buildSnapshot now clamps initialSeconds and derives isCompleted from the clamped total.
• Correct the react entry that listed reset(value) as newly-throwing — only setSeconds is new in 0.3.0 (reset/construction already threw in 0.2.0).
• Set the [0.3.0] date to 2026-06-27.

API docs & examples

• Fix the broken deterministic examples (examples.md, api-reference.md) that asserted on getSnapshot() without driving the engine's interval. They now use fake timers (vi.advanceTimersByTime) so the snapshot refreshes before each assertion. Proven with a real red→green test (old snippet fails expected 2±0 but received 5; fixed snippet passes).
• Remove the false claim that TimeProvider is a public export of @timekeeper-countdown/core (it is internal); point to toTimeProvider() from …/testing-utils.
• Correct the high-level Countdown methods documented as returning boolean — they return void (core-usage.md, core README). The boolean transition result is on the low-level CountdownEngine.
• State that only react (>=17.0.0) is a peer dependency, not react-dom (faq.md, react-integration.md).
• Fix the stale hash-router API Reference link in the react README (/#/api-reference → /api-reference).

Build fix

• Remove a leaked tool-call artifact (</content></invoke>) at the end of docs/changelog.md that broke npm run docs:build (and therefore the docs deploy). Pre-existing on main; scanned all .md for similar artifacts — none other found.

Verification

• npm run docs:build (VitePress) — passes (failed before the artifact removal).
• bin/quality-gate.sh --full — green (build + format:check + lint + typecheck + test; core 420, react 18).
• The corrected example is proven by an actual test (red→green), not asserted.