Does still it matter to be the "first author" nowadays
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.