The car would not shake during normal idle when the car is in neutral or park with one bad mount. But, the engine would visibly move when revved in neutral. You can have someone rev the engine while you are looking at the engine. See if it doesn’t move a few inches or more while the engine is revved, in neutral, drive and reverse.
It’s true there are more than a few reason why your car will shake in drive at a red light but will be smooth in idle. But you didn’t mention very low idle or stalling while in drive at red lights. Does the car also stall at red lights in drive? Is the idle very low?
There aren’t hundreds of reasons why your car will shake violently in drive when stopped but will not shake at all in neutral. There are very few reasons that will not effect idle and will not cause the car to stall in drive while stopped.