So the mission president holds the key to convert baptisms in the sense of approving all convert baptisms.
But the baptism service held in the ward is under the presiding authority of the geographical key holder. As is typical in the church the key holder is the stake president who sanctions bishops in his absence to be the acting presiding high priest/Melchizedek priesthood in their respective areas. Hence when a stake president shows up in an area he automatically becomes the presiding authority.
The mission president is presides of the missionaries and their work and therefore presided over convert baptisms as in people but the Stake President and Bishop by proxy in stake President absence presided over the meetings in that area including baptism services.
