Ok, I found a REALLY cute sample invitation online that you may want to use! See what you think.
There's a Birthday Party
and you're invited to show
But don't tell Chris
because he doesn't know
So if you can make it
here's a word to the wise
make sure you come early
'cuz it's a surprise!
Join us on Sunday, May 4, 2018
at 2:30 p.m.
The Chapter House
256 Lucas Lane
Howell, New Jersey
You can then add information about the attire (formal, casual) or anything else you think may require clarification. I am not sure how people would take the word "invitation", when it comes to paying for the meal. Every person who has invited me to a restaurant for a dinner party has expected the guests to pay, and we all do so, without issue. I have never been to a party where I was off the hook, unless I was the birthday girl herself. I have always assumed that if I wasn't told "food will be provided" when hosting a party at a restaurant, that it meant I paid my way.
Obviously, the "pay your way" thing is only the case for parties hosted in a restaurant. It is understood that in the case of parties hosted in a park or a home, food is provided, usually in the form of finger foods (chips, salsa), catering, or pizza. Perhaps you cay say on your invitation “This is a no Host party – Visit www.(restaurant-name).com for menu and pricing" or “for menu and pricing information, please visit www.(restaurant-name).com” so people don't assume you're spending hundreds of dollars for all of them to eat. It will save you from misunderstandings or awkwardness. I would bring a cake to the restaurant though. That's usually what is done at such parties, rather than forcing guests to buy dessert to celebrate...