How do I invite people to a restaurant, but make it clear that they will each pay their own way?
It’s all in the phrasing. You need to make it clear that you are the coordinator, not the host. Don’t send invitations; rather call and say, “John, would you and Ellen like to meet us at Chez Pierre on Saturday night? We are asking Michelle and Eric, too. We though it would be fun for the six of us to have a night out together. Just let me know if you can come and I’ll make the reservation.” By posing your request this way, it’s clear that you are just the social coordinator and not the host, so everyone should understand that they are paying for their own meal.
However, if you had said, “We’re hosting a dinner at Chez Pierre,” or sent a written invitation with that wording, then your guests would be right to assume that dinner is on you. Once you issue an invitation, you are the host, which means you are paying, even at a restaurant. There is no good way to send an invitation with a note that guests need to be prepared to foot their own bill.