Improve neon init onboarding and preserve fields on .neon writes

[philip]

Update

Split this up as requested. The three independent pieces are now their own PRs off main:

• Install the neon CLI during init, not the retired neonctl alias #432 install the neon CLI, not neonctl. This is the isolated one to merge now.
• Preserve unrelated fields when writing the .neon context file #433 preserve unrelated fields on .neon writes. The non-destructive context write.
• Point neon init doc links at canonical neon.com/docs pages #434 point init's doc links at canonical neon.com/docs pages.

Each is single-purpose and reviewable on its own. #432 and #433 have no dependencies; #434 is for doc URLs. Setting #431 to Draft.

What remains here is the fourth piece: rewiring getting-started to delegate to neon link (plus the neon psql connection check and installing deps before env pull). I stopped here as it depends on #433 for correctness and overlaps with the minimal-init redesign in #428.

Overview

neon init had drifted from the current CLI. It probed for the retired
neonctl alias, so on machines with the neon CLI installed it failed to
detect it and tried to reinstall; it hand-edited .neon (missing the branch)
rather than using neon link; and it referenced doc pages that had since moved.

This reworks the standard onboarding flow to detect and install neon, delegate
org/project selection and the .neon write to neon link --agent (which records
org, project, and branch), point the doc references at current pages, and suggest
neon psql as a quick connection check alongside the driver-based verification.
Dependencies are now installed before linking, so any env pull can safely import a
project neon.ts. The --preview path is left as is.

Separately, and more broadly than init: writing .neon through link, checkout,
or set-context now preserves fields it does not own, so a single neon link no
longer overwrites init's in-progress state. neon link --clear still resets the file.

Verified end-to-end against a live backend: the rewritten init e2e test drives
neon link through project creation, confirms .neon gets org + project + branch,
and that env pull writes a real DATABASE_URL.

Follow-up (not in this PR)

Making init fully backend-friendly is deferred: dropping the --preview gating so
the bootstrap path works for everyone once the platform features are GA, and adding
a plain-Postgres bootstrap template (today's starters all require Functions, Object
Storage, or the AI Gateway). That template also needs a matching example added to
neondatabase/examples. Relatedly, init could scaffold a neon.ts, and the services
it declares under defineConfig's preview field should work without that preview
grouping once GA.

The db and status init phases look like leftovers: the orchestrator routes
onboarding to getting-started and never to them, and nothing in the repo chains to
status, yet they still carry the old hand-rolled project-creation pattern. Pending
a check that no external skill references the db step, they are candidates for
removal.

This pull request and its description were written by Isaac.

[philip]

I created this before seeing #428 which changes things. Still, if 428 is merged, there are bits here worth merging (e.g., neonctl->neon and the link fixes).