Not sure why the definite article is used sometimes after ειναι and sometimes not.
It doesn't have anything to do with "είναι". Here we have a possessive pronoun and the Greek phrasing requires the definite article as well.
What is the difference between "κομπιούτερ" and "υπολογιστής"? and shouldn't this question be in the objects category not the clothes one?
Κομπιούτερ is a trans-literated word. It is the word computer...simply written in Greek letters. Like μπολ, τσίζκεικ and γκρέιπφρουτ.