[release-23.0] Switch to push from pull_request_target in backport.yml (#20093)

[vitess-bot]

Description

This is a backport of #20093

[Copilot]

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[vitess-bot]

Hello @mhamza15, there are conflicts in this backport.

Please address them in order to merge this Pull Request. You can execute the snippet below to reset your branch and resolve the conflict manually.

Make sure you replace origin by the name of the vitessio/vitess remote

git fetch --all
gh pr checkout 20102
git reset --hard origin/release-23.0
git cherry-pick -m 1 d3e04969f088dd395c69ee38e44147b2be489cc5

[mhamza15]

Test backport