This dfa accepts abba as well. I know this is correct as this is from reputed source, but I can't wrap my head around how is this correct. Please guide me.

Trying to provide a hint*, assuming I understand the problem:
-It may help to look at different parts of the DFA in isolation and what each part contribute.
-You can try to apply one specific example to the DFA and check want comes out.

Let's try the second case using "abba". As "a" is entered as input to the DFA, the upper path will be followed. Can you figure out what happens for the next character in "abba", the first "b"? Do we progress to a new state or stay in the same state?

 

*) no complete answer since this is homework section