[wp-trac] [WordPress Trac] #56882: Backport several GitHub Action compatibility fixes

[WordPress Trac]

#56882: Backport several GitHub Action compatibility fixes
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 Reporter:  desrosj           |      Owner:  desrosj
     Type:  task (blessed)    |     Status:  assigned
 Priority:  normal            |  Milestone:  6.2
Component:  Build/Test Tools  |    Version:
 Severity:  normal            |   Keywords:
  Focuses:                    |
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 There are several changes on the GitHub Action platform that need to be
 backported to older branches to ensure workflows continue to run.

 These changes address the following from upstream:
 - Node.js 12 has been deprecated on GHA runners and will be removed in
 2023 (see [https://github.blog/changelog/2022-09-22-github-actions-all-
 actions-will-begin-running-on-node16-instead-of-node12/ upstream blog
 post]).
 - `set-output` and `save-state` have been deprecated. Starting 1st June
 2023 workflows using these commands will fail (see
 [https://github.blog/changelog/2022-10-11-github-actions-deprecating-save-
 state-and-set-output-commands/ upstream blog post]).

 While in some cases, these are not in use in Core workflow YML files,
 third party actions utilize these and need to be updated.

 Related: #56820.

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Ticket URL: <https://core.trac.wordpress.org/ticket/56882>
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