Will carbon fork make a huge difference when commuting? I am considering a hybrid commuter bike, in particular I like Specialized Sirrus Elite Disc. This model however does not sport a carbon fiber fork. Only the more expensive model does, which I do not have budget for.

Another candidate would be Trek Crossrip, which does have carbon fork. Should I make getting carbon fork a priority, will it make a real difference on my ~13 km commute? I sometimes do ride over surfaces like this, that's why I'm concerned:
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.