The Uno getting warm between the DC socket and the USB socket is normal when on external power.
About 0.33watt is dissipated in the regulator when it’s dropping 12volt to 5volt to power the Arduino.
When the Uno is powered through USB, the 5volt regulator is not used (not heating up).
The regulator should not get too hot to touch, assuming you don’t power anything else from the Arduino than the shield.
There is also a <=1Amp diode inline with the DC socket, that can get hot from motor current.
Therefore it’s wise to power the motor shield on the terminal block (bypassing that diode).
- power the setup on the DC socket or terminal block, with the jumper in place (motor power and Arduino power), and accept some heat on the bottom of the Uno.
- power the setup on the terminal block, with the jumper removed (motor power only), and power the Arduino from USB, or a 5volt phone charger connected to the USB socket, or with a lower voltage (9volt) on the DC socket.