Will carbon fork make a huge difference when commuting?
Not much difference in ride from carbon to Cr-Mo.  Over 13km not much of a factor. The carbon is going to be lighter, more expensive, and less durable.  Same bike on a budget I would go Cr-Mo.  You list two different styles of bikes and a purebred road bike in a comment.  Pick the style of bike first.  I don't think purebred road bike is the right bike for that picture.  I personally think drop bars are more efficient and comfortable for paved commuting. And on a budget used might be more cost effective.