fix: add context for run-new-ui-tests dependent jobs

[ZachJW34]

User facing changelog

n/a

Additional details

With #22326 merged in, internal PRs were no longer recording to the dashboard since the jobs didn't have the test-runner:cypress-record-key context and thus didn't have MAIN_RECORD_KEY available.

I thought about changing the MAIN_RECORD_KEY to TEST_LAUNCHPAD_RECORD_KEY but there is a fallback to MAIN_RECORD_KEY in the script so I settled on adding the additional context.

Steps to test

Look at a PR from an internal contributor after #22326 was merged, example:

notice the use of --spec which should only be used for external contributor test splitting.

A build from this PR looks like:

This build is now using the --record flag as intended

How has the user experience changed?

PR Tasks

• Have tests been added/updated?
• Has the original issue (or this PR, if no issue exists) been tagged with a release in ZenHub? (user-facing changes only)
• Has a PR for user-facing changes been opened in cypress-documentation?
• Have API changes been updated in the type definitions?

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[flotwig]

@ZachJW34 is this a dupe of #22559 ?

[ZachJW34]

@flotwig didn't see that one... it is indeed a duplicate. I added it in a few extra places so let me check if that's something we need