chore: Revise UI install text

Description

As described in !2314 (comment 2307092793), shorten the text shown to users who need to sign in, and remove a duplicated link.

Extra credit: revise the product names to reflect our new instructions at:

• https://docs.gitlab.com/ee/development/documentation/styleguide/word_list.html#gitlab-saas
• https://docs.gitlab.com/ee/development/documentation/styleguide/word_list.html#gitlab-self-managed

Related Issues

• Related to fix: authenticate through walkthrough signin (!2314 - merged) where the suggestion to shorten the text came up
• Related to [VS Code] Unable to authenticate using 'Sign In... (#1769 - closed)

How has this been tested?

• If src/browser or src/common has been modified, please consider interoperability with the Web IDE. See Running the Extension in WebIDE.
• Consider an end-to-end test for significant new features that aren't covered by integration tests.

Screenshots (if appropriate)

What CHANGELOG entry will this MR create?

• fix: Bug fix fixes - a user-facing issue in production - included in changelog
• feature: New feature - a user-facing change which adds functionality - included in changelog
• BREAKING CHANGE: (fix or feature that would cause existing functionality to change) - should bump major version, mentioned in the changelog
• None - other non-user-facing changes