Won’t they be printed out as well in the Web UI?

They would in the log, but it will not be stored back on the Task (the idea is these are "agent specific" additions no need for them to go with the Task)

So I’ve tried the approach and it does work,

ScantChimpanzee51 What do you mean it does not work? what exatcly are you trying with task.connect and does not work?

Is there a way to inject environment variables into a Task or into its container?

Yes you can with:
task.set_base_docker(docker_image="mycontainer", docker_arguments="-e KEY=VAL")what am I missing?