Fix build pipeline: Rename service connection from dotnet -> public and update public pool name

[MattGal]

We are excited to review your PR.

So we can do the best job, please check:

• There's a descriptive title that will make sense to other developers some time from now.
• There's associated issues. All PR's should have issue(s) associated - unless a trivial self-evident change such as fixing a typo. You can use the format Fixes #nnnn in your description to cause GitHub to automatically close the issue(s) when your PR is merged.
• Your change description explains what the change does, why you chose your approach, and anything else that reviewers should know.
• You have included any necessary tests in the same PR.

[azure-pipelines]

Commenter does not have sufficient privileges for PR 1115 in repo dotnet/spark

[AFFogarty]

LGTM, thanks @MattGal!

[suhsteve]

LGTM.