From these three photographs, which style of dressing do you identify with most?
I prefer the businesslike style for work, and casual for my free time.
I think you only needed the definite article here.
I think what you have written is similar to what would be written by most natives.
I do not understand why it is suggested that one style requires an article whilst the other dosn't.
Maybe I would say...
I prefer businesslike styles for work and casual clothes for my free time..