DFA/NFA/ε-NFA: subsetting each other or different sets?
If I understand your question correctly, you are essentially talking about the syntactic difference between DFAs, NFAs and NFAs with $\epsilon$ transitions.

Well, in a way, every DFA is also an NFA, and every NFA is also an NFA with $\epsilon$ transitions (it just so happens that the former does not use these transitions).
A caveat here is that if you really stick to the standard definitions, then in a DFA the "type" of the transition function returns a state, whereas in an NFA it returns a subset. 
This is just a technicality, of course, as you can define a DFA to be an NFA whose transition function only returns singletons.

I would look at things this way:
An $\epsilon$-NFA would be the "standard".
Then, define an NFA to be an $\epsilon$-NFA that does not use the $\epsilon$ transitions.
Finally, define a DFA to be an NFA with a single initial state and a transition function that only returns singletons.

In the comments, you ask about the size of the different models, w.r.t a fixed language. This is an entirely different issue. As it turns out, $\epsilon$ transitions can always be removed, and no new states are introduced in the process, only new transitions.
As for the translation to DFA - this may involve an exponential blowup in the state space.