I know you have already got a wealth of advice, but wanted to give my unique point of view.
I was a nanny for ten years before having my daughter, now 22 months old. So I picked up a few tips along the way that are working great with our daughter.
When we are at home we are very specific in what we expect of her at meal times. She MUST sit in her seat and we encourage her to be independent and feed herself, but we also include her in our conversation. If my husband brought his laptop to the dinner table that would drive me crazy and teaching our daughter that she needs to have a toy at the table only encourages the idea that she is there to play, not to eat.
When we eat out, the same rules apply. If she screams we quietly, but firmly tell her that we don't do that and engage her in conversation. If she thinks we playing then sometimes reverse psychology worked on our little one at your daughter's age. We would turn away and say "We don't talk to little girl's who don't listen to their Mama and Daddy." This turn of the attention away from her would sometimes get her attention. If she screams again, pick up a book and start reading it. This should redirect her attention. When she stops screaming and looks at the book make a big deal of thanking her for not screaming. Remind her what a good girl she is when she's not screaming.
This may take a few tries but you have to teach her how to behave in restaurants now or you'll be eating at lightening speed when she's older because she's got ants in her pants.
I would recommend only letting her out of her seat if you are taking her to the bathroom to 'wash her hands' before the food arrives. If this trip from the table has a purpose other than getting out of her seat then she will still learn that you stay in your seat.
Most important when you sit down at a restaurant is to make sure that she's not really hungry as you're getting drinks ordered and looking over the menu. Making sure she has something already to eat can help her disposition.
She is old enough to understand if you tell her each time she wants to get up that Mommy and Daddy aren't done eating yet, so she needs to be patient. The concept is something she will grasp more the more you do it.
I hope this helps. Good luck Mama!
S. M
Stork's Best Friend
www.storksbestfriend.com