You can think of zero point energy as the energy in an empty region of space. Of course this region isn't totally empty or else there would be zero energy there. The zero point energy is the contribution from virtual particles predicted by quantum field theory. Some of the members who are better versed in QFT can probably elaborate on this point if you are interested. Cosmologically, the zero point energy is the most straight forward explanation for dark energy (I use straight forward liberally here).
Is another ame for the zero point energy field, the vacuum?
from a scholasticism type analysis , vacuum should not be claimed to be "with out anything", because for "without anything" we get a unique concept of vacuum. But it is not vacuum is not "one" concept. we can have a space time extension in which more than one quantum field exist , one field A have all it's mode in vacuum state but the other field B do have some excited modes. So the spacetime area is vacuum for A but not for B.
generally we have many vacuums, I think. And in the calculation of vacuum energy for an dark energy explanation ,we just take into account the vacuum of one scalar field ,which is just hahaha ,assumed, to be exist.