It seems that there are two kinds of vibration:
a) vibration due to faulty brake mechanism and
b) vibration due to engine rpm below normal value (say 650 rpm).
(a) is common to both MT and AT. AT cars require a harder brake effort because there is less engine braking effect.
(b) is common specially to AT.
For case of AT. Before warming up, the ECU will set to idle engine rpm to over 1000 rpm, there is no chance for the engine to drop below 650 rpm and cause the engine vibration.
After warning up, the ECU will set idle engine speed at about 650 rpm. There should be no engine vibration if the gear in Neutral. But if it is in gear, the engine will tend to run slightly lower.
But the engine rpm will be much lower under the following conditions:
1) When the car is braking, the ECU will cut off the fuel if the engine rpm is above 1000 rpm.
2) When the car is nearly stopped, the ECU will tend to supply back the fuel to maintain the normal idle rpm (650 rpm). But the initial fuel supply may not be enough and the engine rpm falls lower than 650 rpm. If so, the engine vibration would occur until the ECU responses to supply a little bit more fuel to raise the rpm back to normal.
At nearly stopping, the engine is practically de-coupled to the wheels. However, the AT at stopping in D will still give a slight load to the engine.
For MT at stopping, there is practically no load to the engine if the clutch is de-pressed and engine vibration is rarely occurred.
I notice that the ECU will raise the idle rpm to 750-900 rpm for any of the followings:
a) the parking lights are ON;
b) the headlights are ON;
c) the air-con compressor is ON due to de-miser and cooling.
Do you notice these?