If it's 8-12 inches in diameter, round, has shallow sides and has a handle - it's called a round griddle or a griddle pan. Most that I have seen don't come with a trivet for table service - but a few do. Emeril has one that has a little taller sides that he calls a Quesadilla pan (but no trivet). And, Lodge makes what they call a round serving griddle
(griddle without a handle) - and they have an optional trivet you can buy for it that would probably work with any other similar sized round griddle pan. Heck - you can get an el-cheapo round cuttting board that will work just as well.
If you want ovel, with a side about an inch high, and a trivet - these were common in steakhouses before the fajita craze. Depending on the manufacturer - these are called sizzle/sizzling/sizzler pans/plates/platters - the term fajita, or steak, may or may not be added to the name. These come with trivets - generally wood but a few come with a wicker trivet
Some are made with no handles
, little stubby "pot" type handles
, some have a straight pan type handle
, some are bent at a 45º angle (like the one with the willow trivet above), and some are bent at an almost 90º angle
They also make these out of aluminum - but I prefer the ones made from cast iron because they stay hotter longer.
Just Google the different terms/combinations of terms and find what you want at the best price. Also, if you have a local restaurant supply - they might be able to help you if you use the term "sizzle plate".
Hope this helps you find exactly what you're looking for.