Resolve regressions from proxy-output fix

[scbedd]

Hey @JoshLove-msft I'm mostly free of any fires, so I have some time to apply to this.

First, some context. I resolved an issue with the proxy not properly outputting to stderr when using logging.error. However, when I resolved that issue, the next eng/common update actually broke a few additional test cases, because they were checking at the end for stderr output...and suddenly were getting some! Here is your comment on that sync-common PR at the time.

I'm filing this PR now because I have the time to apply fixes to azure-sdk-for-net, and I don't want to maintain a separate version of the proxy that doesn't output to stderr!

First, let's see what melts here.

[github-actions]

Hi @scbedd. Thank you for your interest in helping to improve the Azure SDK experience and for your contribution. We've noticed that there hasn't been recent engagement on this pull request. If this is still an active work stream, please let us know by pushing some changes or leaving a comment. Otherwise, we'll close this out in 7 days.

[github-actions]

Hi @scbedd. Thank you for your contribution. Since there hasn't been recent engagement, we're going to close this out. Feel free to respond with a comment containing /reopen if you'd like to continue working on these changes. Please be sure to use the command to reopen or remove the no-recent-activity label; otherwise, this is likely to be closed again with the next cleanup pass.