Welcome to Drum Chat GinkoSan!
First of all, the 2 techniques for pedal work are referred to as "heel down," or "heel up." Your teacher is having you work on the heel down technique. What you describe as lifting your whole foot and dropping it toe first is a description of the heel up technique. You usually get more power, and more endurance with the heel up technique. Your explanation is pretty good, but I would like to know what pedal you are using for the "bass drum." I believe that what you are calling the stick hanging out of the pedal is probably the beater. This can be adjusted, as can other things on the pedal.
If you can tell us what pedal you have, or if you can post a picture of it, maybe we can give you more specific advise.
Again, welcome to Drum Chat!
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