You could add a step to the pipeline itself that does the pull manually, I guess. If I google around, I see more people are annoyed that this information is going to stderr (for no visible good reason why it is). Docker is a very weird CLI tool, with many quirks, sadly.

Another approach might be to figure out how to show warnings/errors like the MSBuild can. I would assume there is some way to do that. I just haven't found it.

Possibly you could ask on the azure-pipelines github issue tracker for help. Maybe others have suggestions and ideas.

I know we don't use Docker@2 task, probably because it doesn't have run command, but in microsoft/azure-pipelines-tasks#11118 it's said a script is enough to run a docker, and maybe that could be a solution to filter/clean the output

The new and shiny way of doing this is GitHub Actions, which has solved these kind of things a bit smoother. The OpenTTD account is signed up, so it might be worth a look.