Update deploy-staging to trigger only after successful test run instead of test completion

[clairep94]

Issue:

Fixes #
Part 1 of the remake of #4252

• Currently deploy-staging always gets triggered upon unit test completion, even if a test run has failed
• This PR updates to only trigger deploy-staging if the unit tests pass

Demo:

Changes:

I have verified that this pull request:

• has no linting errors (npm run lint)
• has no test errors (npm run test)
• has no typecheck errors (npm run typecheck)
• is from a uniquely-named feature branch and is up to date with the develop branch.
• is descriptively named and links to an issue number, i.e. Fixes #123
• meets the standards outlined in the accessibility guidelines