Please listen to me, I'm a drift instructor in real life and can see all your mistakes in the way 90% of my students start out.
Your biggest problem is that you are starting off with bucketloads of understeer. This will cause you several problems.
1) When drifting, we rely on the castor of the front wheels to 'self-centre' and correct the slide (the reason we let go of the wheel when initiating a drift). If you start with understeer, the front tyres are already sliding and lose their desire to 'self-centre' and give no 'feeling'. This is why the car immediately spins 9 times out of 10 (one of the exceptions being at 4:05-ish)
2) In order to initiate a drift, you require the front wheels to grip and the rear wheels to slip, make sure you have enough grip to turn in with in the first place, then break traction at the rear. You current technique involves turning in way to fast, which causes understeer, which means that when the rear does then break away, it's uncontrollable.
People spend all their time associating drifting with power and wheelspin, but the simple fact is that the PRIMARY method of controlling and initiating a drift is the steering.
The best tip I can give is to watch your replays from outside the car.
Your front wheels should ALWAYS point in the desired direction of travel (except in a four wheel slip/backwards entry scenario, but that's an advanced technique, so not relevant to you yet
). If they aren't doing that, you are going to either make a mistake, or worse, go off track.
Start by entering the corner slowly, turn in smoothly (and slightly late) and then apply the throttle smoothly and firmly. As the rear slides, you need to apply lock at an equal rate (achieved by letting go of the wheel as the rear breaks away) and then using the throttle to control the slide. As said, before, you will be better off using a higher gear. i.e. if you would use 2nd gear when racing round a corner, use 3rd when trying to drift it.
Have a go, try a few things and then get back to me. I can try to explain it slightly better, or maybe try to post a video of GT6 vs real life (which is SHOCKINGLY similar).