This question is based off An example of non-empty regular language for which a DFA with single accepting state doesn't exist.
Let $C$-type language be a regular language iff exists a deterministic finite automaton (DFA) $A$ which has a single accepting state such that $L=L(A)$. An example of a regular non-empty language which is not of $C$-type is $L=\{1,10\}$.
I don't understand what single accepting state means. According to my intuition the DFA for this example would look like:
Indeed there's a single accepting state here (double circle) so I don't understand why this DFA doesn't have a single accepting state.
To turn the comment into an answer:
The automaton you drew does indeed only have a single accepting state, but the language of the automaton is $10^*$, not $\{1,10\}$. For example, your automaton accepts $100$, which is not in $\{1,10\}$.