It can be a bit confusing at first but read the above post. No need to "re-apply" for a third class medical to begin your flight training. Look at it backwards: You can use a first class medical when a third or second class is required but not the other way around. So for you, specifically, if you get approved for a first class medical and use that for your flight training, think of it as having a third class medical for 24 months.

The reason for the advice to obtain a first class medical initially is because that will be required for the airlines (part 121) and if you cannot obtain it for some reason, you learn that before shelling out $50k+ for flight training. Since the medical "degrades" so to speak, (first into a second, second into a third), without the need for the pilot to do anything, it works for all portions of flight training.