I'm very curious which languages most people asking about this question speak. In English, indeed, the phrase "(not) allowed" is completely ambiguous and context based! Maybe kind of tense-based as well -- present tense is usually about permission and policy, and past or future tense implies more of an active role.

"I don't allow my child to watch TV" - implies that I have a policy which forbids it, but the child might sometimes turn it on if I'm in the other room.

"I didn't allow him to watch TV that day" - implies that I was completely successful in preventing him from watching TV.

"I won't allow him to watch TV on the airplane" - implies that I plan to fully prevent it.

"My company doesn't allow any non-company-provided software to be installed on our company computers" - totally ambiguous. Could be a pure verbal policy with honor-system or just monitoring, or could be fully impossible to do.

yes, exactly. It has permissions to send email, but it is told to not to send emails with human approval.

Yes hopefully this is the case. I'd prefer if it were worded more like:

He has access to reply but has been told not to reply without human approval.