The person the spell was cast on pays the Downtime
The phrasing of the answer in the AL FAQ makes this very clear
[...] until the recipient spends a total of 120 downtime days [...]
Let me clarify the phrasing a little bit. You seem to be reading this as if this FAQ answer was directed to everyone. But, in fact, the answer is written as if speaking to a spellcaster who is capable of casting Clone.
To rephrase in a way that is less succinct, but more clear...this is essentially what they are saying:
"If you cast the clone spell on someone (yourself or another), then the clone is not mature until the person the spell was cast on spends a total of 120 downtime days after you cast the spell on that individual."
To rephrase it for you, as someone on whom the spell is cast...
"If someone else casts the Clone spell on you, then the clone is not mature until you have spent a total of 120 downtime days after they cast the spell."
Your confusion simply stems from who the subject of the answer was.