Officers arriving on the scene found Terra critically injured in the rectory of the church, Phoenix Police Sgt. Steve Martos told NBC News early Thursday.
Terra — who first called 911 to report a burglary — was transported to an area hospital where he remained in critical but stable condition, police added. Officials at the news conference declined to detail what caused his injuries, but said they plan to interview him again when he's ready.
Officers at the scene had located Walker suffering from a gunshot wound, and he later died at a local hospital.
The priests were members of The Priestly Fraternity of St. Peter, which offers traditional Latin masses to its congregants.
Walker was ordained as a priest in May 2012, and was known in Phoenix as a "very faithful servant," Adamson said.
Fr. Michael Stinson, of The Priestly Fraternity of St. Peter, remembered him as a "young, outgoing priest," and told NBC News the community has been "in shock" since Wednesday night.
Although police were called to the church for a robbery, they declined to say if anything from inside was taken.
One of the victim's vehicles was initially reported missing but later discovered a few blocks away from the crime scene, Martos said.
At the news conference, Adamson said the priests had thought of the neighborhood as a "safe place to live."
"The attack that occurred last night at the Mother of Mercy Mission Church strikes at the core because every American should feel safe in a house of worship," added Phoenix Mayor Greg Stanton. "That basic value was violated last night."