Languages accepted by modified versions of finite automata
This combined with Alex's answer give the complete picture.

$L(DDFA)\subseteq L(DFA)$ can be proven by adapting the usual powerset construction with a modified final state condition. In the power set construction, states are sets of states from the original automaton. Usually after performing the powerset construction, a state is final if one of the states in the set is final in the original automaton. 


In type-1 DDFA, final states in the constructed automaton are the sets where all elements are final in the original automaton. 
In type-2 DDFA, final states are the singleton sets of the final states from the original automaton.


In both cases, the resulting automata are DFAs.

Now type-2DDFA's can express only the languages $\{\epsilon\}$ and $\emptyset$, depending on whether the starting state is accepting or not. This is because the two transitions from a state need to go to distinct states, but acceptance is only possible if they end up on the same state.