I'll presage this by saying that you're going to create problems for yourself by thinking too hard about this.
First, what type of data is my DV?. It might seems like categorical, but the data is more than just like gender of male or female. It has values of intelligent and accuracy in it.
Your intuition is correct, although your reasoning is not entirely sound.
The DV is dichotomous (binary), but it can also be viewed as an ordinal variable, because here, the value of 1 refers to the respondent possessing more of something. In this case, 1 means they have more knowledge/ability/proficiency.
data is more than just like gender of male or female.
Says who? Gender is traditionally viewed as a categorical variable, but if we define gender as the number of penises present, it's actually an interval scale.
Second question is, which statistical measurement should I use?.
You would need to elaborate more on what your study is. For example, for reasons noted above, personality type could be viewed as an ordinal scale where an individual increases in competitiveness or propensity to give themselves a heart attack (type Bii, Bi, Aii, Ai -- or whatever archaic personality theory you're going for)
Most likely, personality is a categorical variable and the study has a 2x4 design. In which case you'd perform a chi-square test.