You are correct with this assessment. But the way the benefits are dispensed are different and thus you can use Uber cash with the Amex benefit but not with the Chase Lyft benefit.

Amex benefits are applied to your account as Uber cash albeit they have separate buckets for different "types" of Uber cash. Thus you obviously must have Uber cash toggled on to be able to have that money applied to your trip, however you must also have an Amex card selected as the backup payment method. Now, if you have other Uber cash in your account on addition to the ones from Amex, that will be used for any overage before your card actually gets charged.

With Lyft the $10 is applied as a discount when you are requesting the ride. If you have Lyft cash toggled on, even if your CSR is selected as the backup method, the $10 discount won't show up for that ride.