There are multiple ways to order an abstract. I work for a trucking company and order them all the time. Make sure that when you order you request an abstract that goes back as far as they allow. I do not know what Uber orders but in some states a standard only has 3 years and in some states it is lifetime. Other states will give you 3 but more years are available.
First I would say it doesn't hurt to send them an application to drive. You have to pay to get those records and it doesn't sound like it would be a problem so why bother? They look at your points, was your license suspended and in this case it doesn't sound like its an issue.