I was looking for definite conditions for a DFA to be universal.
So far, I've mainly found this:
A DFA is universal iff all its states (minimal states) are final.
This does not seem enough however: a DFA having a single state, which would be initial and final at the same time, will not be universal and would only accept the empty language, should there be no transitions over that state.
What is the correct/formal criterion in order for the DFA to be universal?
The statement "A DFA is universal iff all its states are final" is only true, if according to your definition a DFA has to be complete (which indeed is required often). But your example automaton is not complete, because it does not have transitions for all symbols from its unique state. If you add these, the automaton indeed accepts $\Sigma^*$ and is thus universal.
To be more explicit, you could formulate your condition as:
This makes sense, because in a complete DFA every string is processed along some path, which necessarily ends in some state; if all states are final, all strings are accepted.