Icons: Allow registering icons that are hidden from the REST API

[fushar]

To support replacing Dashicons with SVG icons in admin screens via #65089, we need a way to register an icon which is available via wp_get_icon() but not available via REST API and not exposed in the Icon block.

The current consensus of the approach has been discussed here: ​https://github.com/WordPress/gutenberg/pull/79451#issuecomment-4797981729. In short, we propose to allow a new property public when registering an icon via wp_register_icon().

Here's a summary of the behavior:

Registered with	Registry / wp_get_icon()	REST API / Icon block
public omitted or true	yes	yes
public: false	yes	no

This Trac ticket is also tracking Core backport for the Gutenberg-side changes here: ​https://github.com/WordPress/gutenberg/pull/82634.

Trac ticket: https://core.trac.wordpress.org/ticket/66087.
Backport from ​https://github.com/WordPress/gutenberg/pull/82634.

## What

This PR allows an optional boolean public property to icon registration via wp_register_icon().

Icons registered with public: false will still be available through wp_get_icon(), but the REST API will not return them, so they aren't exposed in the Icon block.

As a back-compatibility, if we don't provide a public property, the default is true and the behavior will be the same as the existing behavior.

Here's a summary of the behavior:

Registered with	Registry /wp_get_icon()	REST API / Icon block
public omitted or true	yes	yes
public: false	yes	no

## Use of AI Tools

Used Claude Opus to generate the backported changes from the linked Gutenberg PR, then checked the changes myself.