As a German I would definitely use "anrufen" when using Skype or similar systems.

Calling on the phone or modern equivalents (voice calls on skype/whatsapp/vidyo/...) is the most common meaning, the word itself is older and has the meaning of calling someone by his or her name. It is also used to call for help/advice in certain ways, like ein Gericht anrufen ("calling a court") or praying for something ("Gott anrufen"). In radio communication (ships/planes/amateur radio) you say rufen, like already stated in the star trek example.

Rufen can. :)

Yeah, why not?

Not only can you use "anrufen", that's actually the only usage where this word is mostly still used in German today. Although technically you can use it for calling someone "in real life" (i.e. not on the telephone), Germans mostly use just "rufen" for that.