I call the second method, the one your coach told you, the "armadillo method". Basically, you put your head down on the mat, shell up, seal off, and don't let him get any control on you. Then, you can work on your stand ups and other moves to escape. National finalist and long time UNC head coach Bill Lam preached this method, so obviously it works. Is it a bad method? No. I wouldn't call it better than the other (immediate action). But both are valid strategies.
As far as my opinion, I've always felt that bottom is about creating motion and being able to chain together moves in such a way that your opponent can't follow you. If you're sitting there with your head and weight on the mat, you're obviously not creating any motion. And while you're sitting there, even though you've sealed off and he can't get inside, all of his weight is still on top of you. I feel like it's very hard to create that motion and come up after he's already had time to put all of his pressure and weight on top of you. Instead, why let him have the chance to do that? Stand-up off the whistle while sealing off at the same time, get your hand control, cut away. If the stand up doesn't work, chain it into something else. I never like to stop moving on bottom.
I guess that I believe in the theory of constant motion on bottom, and when you're sitting down there letting him get his pressure and weight on top of you, it goes against that. However, it IS a valid tactic. Your coach isn't wrong by any means. I feel that, if you prefer standing up on the whistle, your coach should be fine with that. If you grow to like the armadillo method, then use it.