If you've yet to 'master' either technique, you don't want to by trying to learn both at the same time. As it's all muscle memory, trying to do alternate and economy picking will probably lead you to randomly mixing the two.
Alternate picking is a more useful technique, so that should be the starting point. Once you can alternate pick consistently and continue while skipping strings, you can think about switching from inside to outside picking alternate, and then economy picking. For economy your licks need to be worked out so you have the right number of notes per string so you can continue picking in the same direction as you cross strings.
I don't have a guitar in my hand, so am having to visualise, but I don't think it's possible to truly economy pick the run you've tabbed. To be able to economy pick it by my reckoning you'd need to take another note on the E string when you drop back down to it (4 instead of 3) otherwise your pick is heading in the wrong direction and you're alt picking anyway.
I teach people to stick to one until they have it down, and then look at the others. It's like if you tried to learn a whole/half diminished scale at the same time as the half/whole - you'd probably end up playing chromatics...
I think your teacher is right to tell you to stick to one for now - you can't build a solid picking technique if your hands change styles without you realising.... :eek:
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