This looks like a power issue. The MG90D might consume a bit more power than the fake ones.
Also, note that if you're using digital servos you should change the robot code to use a higher frequency for the servo signals. Around line 130 you'll find this code:
int FreqMult = 1; // PWM frequency multiplier, use 1 for analog servos and up to about 3 for digital.
// The recommended setting for digital is 2 (probably safe for all digital servos)
// A shunt between Nano D5 and D6 will set this to "1" in setup, this allows you
// to select digital servo mode (2) or analog servo mode (1) using a shunt
// or short jumper wire.
If you set FreqMult to 2 you will likely have better luck with digital servos because they like running at higher frequency than analog. You can also put a shunt between D5 and D6 to enable this mode (shunt must be in place on boot). You could rig up some wires and route them to the cap area so you could plug and unplug this shunt at will and directly test the differnce without pulling the nano out.
However, the symptoms shown in your video seem too severe for that to totally fix it. But it's worth a shot.
It's possible your batteries cannot output enough amps to handle the digital servos. Could you send me a picture of the battery label, being sure to include both sides of the battery?