Does still it matter to be the "first author" nowadays It used to be that the first author (meaning, the author whose name comes before the others) is thought to have contributed most to the publication.

But in the last couple of decades, it seems that the first author no longer matters AT ALL, instead the last person has become the most important. 

I attribute my observation to two things,


The first author is typically a student, someone with no name recognition, or someone who is getting advised or funded by the last author.
The last author is typically the director or advisor of the project. In recent years many of these people have become ultra famous through things like TEDtalks and other media hype. It didn't use to be this way before the internet. 


Nowadays whenever I evaluate how potentially interesting a paper is in my field (stem), before I even read it, I almost always look at the last author, especially if it is someone famous. If I don't recognize anyone, I am quick to assume that the paper is not interesting. I am trying to fight back against my bias. 

Does being the "first author" still carry any weight in today's academic culture?
In fields that don't do alphabetical order being "first author" still matters most in early career stages.

As you correctly say the senior author is usually a professor (= group leader) while the first author is the person that did most of the work. In an early career stage (until you become group leader or assistant professor) first authorships are the most important thing. As soon as you start your own research group the senior authorships become more important (in rare cases some very senior postdocs might have one or the other senior authorship as well).

In many application guidelines you would find instructions like "attach separate lists of (a) all publications, (b) first authorships and (c) senior authorships". If you don't have a sufficient number of first authorships they will not even look at your application in more detail.