To treat vs. to invite

Inspired by a recent stay in a hotel (see picture), let me point out a subtle but important difference in meaning between the verbs treat and invite. The situation is made even trickier by the fact that other languages (in particular German) do not have this distinction. Consider the following two sentences:

(1) I would like to invite you to dinner.

(2) I would like to treat you to dinner.

In English, (1) does not automatically imply that I will pay for your dinner. All you know is that I want your company. In contrast, (2) means I would like to have dinner with you and will pay for your food and drinks. While this may be confusing, in this case, English is actually more precise than, for example, German. In German, the sentence

Ich möchte dich zum Abendessen einladen.

could in principle mean either (1) or (2). Finally, the typical English expression used when the check comes is

It’s my treat.