"Do you begin?"
Thats a really hard question. To be clear i believe "αρχίζεις" is more appropriate translation to "do you begin" than "ξεκινάς" so the guys who made the lessons probably did it to make your life easier.
Simple explanation here :
Ξεκινάω=Αρχίζω,these two words are synonyms in Greek so yeah in this case both are right BUT in most cases it would sound weird to use "αρχίζω" or even change the meaning of a sentence while "ξεκινάω" is always correct.
So if you want more in depth explanation :
Here is an example : I start the car = Ξεκινάω το αυτοκίνητο.
In this case it would be weird to say "Αρχίζω το αυτοκίνητο". In the same manner it would be weird to say in english "i begin the car" wouldnt it? If you say to a Greek "Αρχίζω το αυτοκίνητo" and you are in a car he will laugh because its wrong but he will understand what you mean. If you say it outside of a car its still wrong but he will probably think you are beginning driving lessons.
In another example it would be ok to say this : Αρχισα τα αγγλικα which translates to "I started/began learning english".
In most cases us Greeks use "αρχίζω" as the beginning of a long term process
examples incoming :
i begin the reading of a book=αρχίζω το διάβασμα ενός βιβλίου
I begin driving lessons=Αρχίζω μαθήματα οδήγησης.
So since i am not a teacher i cant explain better the use of "αρχίζω" but i can tell you this, "ξεκινάω" is correct 100% of the time but "αρχίζω" is something that you will learn to use properly when and if you get fluent in Greek. I hope i helped.
Thank you AkisV1.
Since there is no context "Αρχίζεις;" could also be used. It has been added to the translations that are accepted.