Do the slaughtered animals of Jews and Christians become permissible to eat if they pronounce "In the name of the Lord" over them?

The default ruling is that the animal slaughtered by a person of the Book is permissible unless it is known that they invoked a name other than Allah. If a Jew says "in the name of the Lord," there is no issue with the permissibility of their slaughtered animal. If a Christian says "in the name of the Lord," their slaughtered animal is permissible, because they have not outwardly invoked a name other than Allah. Even if the Christian intends by this Christ, or if the slaughtered animal is for a holiday or to draw closer to someone they revere, it is permissible as long as they have invoked Allah’s name alone and have not invoked another name. However, it is disliked due to the difference of opinion concerning it.