chore: update developer guide links in _index.md (#23372)

[ti-chi-bot]

This is an automated cherry-pick of #23372

What is changed, added or deleted? (Required)

Update developer guide links in _index.md to the latest links

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• v6.5 (TiDB 6.5 versions)

What is the related PR or file link(s)?

• This PR is translated from: chore: update developer guide links in _index.md docs-cn#21852
• Other reference link(s):

AI agent involvement

• The changes in this PR were primarily made by an AI agent on behalf of the PR author.

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• Delete files
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• Need modification after applied to another branch
• Might cause conflicts after applied to another branch

Summary by CodeRabbit

• Documentation
  • Updated the Develop learning path with revised developer guide and quick-start links.
  • Added a “Connect to TiDB” resource.
  • Removed the previous “Example Application” link from the Develop section.

[ti-chi-bot]

@qiancai This PR has conflicts, I have hold it.
Please resolve them or ask others to resolve them, then comment /unhold to remove the hold label.

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