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It does sound like "contacted the church" means "we called up the church since we knew we needed a church in order to have a christening." I SUSPECT that if they were regular church attenders they MIGHT say "our church" or even "we spoke to our pastor" or "our priest."
The "upper middle class Episcopalians in the South" who have a "private ceremony with maybe 50 or 75 people in attendance, mostly family and old friends" and then brunch at the country club, are definitely retro: since the adoption of the 1979 Book of Common Prayer, Episcopal baptisms are supposed to be celebrated at the regular Sunday service in the presence of the regular congregation, who, in the ceremony, are called upon to promise that they "will do all in their power to support this child in his/her new life in Christ."
Most churches have been making a real effort in the last 50 years or so to move baptism from a private ceremony to a ceremony focused on the worshiping community, and from an insurance-policy-against-baby-going-to-hell to a celebration of the launching of a life of serious Christian commitment.
Not always successfully.