Deprecate the Link Manager (Bookmarks) API (see #56362)

[georgestephanis]

Related: #56362

Step 1 of that ticket's rollout plan — moving the Link Manager functionality into a plugin defensively — is done: https://github.com/georgestephanis/wp-links reproduces this API under the same function/class names, guarded so it activates only once core stops providing them.

This PR is step 2: flag everything as deprecated, with no behavior change. _deprecated_function(), _deprecated_class(), and _deprecated_file() only emit a notice when WP_DEBUG is on — every function, class, and admin page here keeps working exactly as it does today. This should be safe to merge well ahead of any actual removal, giving plugin/theme authors real advance notice.

What's deprecated

Matches the companion plugin's coverage:

• get_bookmark(), get_bookmark_field(), get_bookmarks(), sanitize_bookmark(), sanitize_bookmark_field(), clean_bookmark_cache()
• wp_list_bookmarks()
• get_edit_bookmark_link(), edit_bookmark_link() (in the otherwise-generic wp-includes/link-template.php, but part of this API surface)
• add_link(), edit_link(), get_default_link_to_edit(), wp_delete_link(), wp_get_link_cats(), get_link_to_edit(), wp_insert_link(), wp_set_link_cats(), wp_update_link()
• wp_link_category_checklist()
• WP_Links_List_Table, WP_Widget_Links (via _deprecated_class() in each constructor)
• wp-admin/link-manager.php, link.php, link-add.php, edit-link-form.php, link-parse-opml.php, wp-links-opml.php (via _deprecated_file())

Every deprecation message points at installing the companion plugin — there's no in-core replacement function to name, since the point is this functionality moving out of core entirely.

Deliberately not touched here

Matches what the companion plugin doesn't cover either — tracked in its issue #1 and issue #2:

• The link_category taxonomy registration and manage_links capability mapping — no established core convention for deprecating a taxonomy or capability at runtime the way there is for functions/classes/files
• The Links menu entries
• The five link_*_meta_box() functions and xfn_check() — internal render callbacks for the now-deprecated edit-link-form.php; deprecating them individually seemed redundant with deprecating that file
• The two Links-specific AJAX handlers
• The wp_links DB table, link.js/xfn.js
• The already-19-years-deprecated stubs in wp-includes/deprecated.php

How this was verified

Not just PHP lint — tested against a real WordPress environment (WP Playground): link category creation, link insertion, get_bookmarks(), wp_list_bookmarks() rendering, and widget instantiation all still succeed exactly as before, while each one correctly emits a PHP_DEPRECATED-level notice under WP_DEBUG.

[github-actions]

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[github-actions]

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[georgestephanis]

Companion preview PR showing what the eventual final removal (step 5 of #56362's plan) would look like on top of this deprecation: #13360