at the time of the Adoption of this Constitution
That adoption was not instantaneous, nor was there time limit in the Constitution itself, so the two acts were passed in order to end that time limit.
No Person except a natural born Citizen, or a Citizen of the United States
The different terms in the two acts were used for the same reason the same exact terms were also used in the grandfather clause - to distinguish between a natural born and naturalized citizen.
so the two acts were passed in order to >begin and< end that time limit.