"One who heard us was a woman named Lydia, from the city of Thyatira, a seller of purple goods, who was a worshiper of God. The Lord opened her heart to pay attention to what was said by Paul. [Acts 16:14]"
Yes, this would show the way it is done by the Holy Spirit, because the HS knows the timing of when a person is ready. We are not born, "born again". However the "decision" is part of when this happens...when a person professes faith. If you want to say that God makes our mouths move to profess faith, that is fine (for you), but most people will say in human terms that they spoke their heart.
Perhaps the word "decision" does not fit a Calvinist ideology very well, but it explains part of the whole process of being born again. If someone does not profess Jesus as their Lord and Savior, would they be saved? The bible says no.