Our current DFA shown below, $M$, accepts binary strings with an even number of 0's and odd number of 1's. (since it only has accept state $q_1$).
Suppose $q_1$ is no longer an accept state and $q_0$ is now added as an accept state. Then we can say the modified DFA M' accepts strings with an even number of 0's and even number of 1's.
What if instead we don't remove $q_1$ and still add $q_0$ as an additional accept state to obtain the DFA $M''$. Can we say $M''$ accepts binary strings with an even number of 0's and an even number of 1's and $M''$ accepts binary strings with an even number of 0's and an odd number of 1's. Or can we just say $M''$ accepts the binary string as long as it has an even number of 0's?
I would like to know how should we generalize our languages to describe some DFA like this ?
If both $q_0$ and $q_1$ are accept states then yes, the DFA will accept a binary string as long it contains an even number of zeroes. The two ways of describing the DFA you've given are the same so it doesn't matter which one you go with (althought the second definition is shorter).