Thanks for asking your question on the forum. As you already found the documentation, I will try to answer your question as brief as possible so you know what you're up to.
- Yes, you do need a license if you want to host a Mendix app on cloud services. If you don't have a license, it will behave comparable to your local environment. The app will shut down after a few hours.
- Yes, you do need a license if you want to host the app on your office server. This counts as on-premises. From the top of my mind, this is the same license.
Hopefully, this answers your question. If so, it would be great if you can mark the question as answered. If not, I am sure that a Mendix CSM can help you out with your specific request.
Free application :
You can deploy your Mendix application without any license. But a free application has a number of limitations compared to a licensed app. You can check this for more details :
Licensed Application :