I don't say "I invite you for dinner". That's not good English. There are better ways to ask. I would be just as confused as you if somebody asked me that.

I want to second Johnny’s comment that “I invite you for dinner” is poor English. There are plenty of things that you can be invited to, though, where you would be expected to bring something or offer to bring something. We’re invited to Jeff’s birthday party (gift) We’re invited to a Memorial Day picnic at the Johnson’s (typically potluck, so bring a dish) The Smiths invited us over for dinner (a bottle of wine, dessert etc) I’d like to invite you over for dinner. I’d like to invite you to dinner. (In a dating situation, these two would imply that the invitee doesn’t have to pay, though it could be nice to offer something)

In the UK if I invite you to dinner, it can mean I invite you to my home for a meal, or I will take you out and I will pay.

If they invited you, then let them pay, but if they say something like *let's go out for dinner that means you might end up sharing the bill*

Same for Switzerland. If you say 'Invite' than it means you pay. Otherwise you would say something like: Let's have dinner together/ How about dinner tonight and so on.