I call this "clean labelling". This is when you do the same things, with a different name. Very often, they will pick a different day, and have presents wrapped in plain white paper instead of the pretty (and sometimes expensive) wrapping paper or gift boxes. Sometimes they will pick a day in the middle of January, after they can get things off price, for this day. Occasionally, there will also be some of the other trappings--eggnog, Easter candy, and turkey. It will never be called Christmas, nor will there be Christmas music or decorations. Occasionally, they will pick out some "seasonal" music or "seasonal" decorations and display those, as well.
In fact, there is a wide disparity of tolerance for this practice within congregations. Some are liberal, even tolerating people driving around to look at the Christmas lights. And they will allow people to sing "winter" songs (you know, those "Christmas" songs that actually glorify winter instead of the holiday itself). Other congregations have zero tolerance--and you will get an all-expense paid trip to the back room for driving around looking at the Christmas lights or singing winter themed songs.
What I predict one day is that the Filthful and Disgraceful Slavebugger is going to get wind of this practice. I am looking for the Washtowel article that cracks down hard on clean label holiday or birthday celebrations. It will feature the illustration of a little poison in your water (the holiday is the "poison"), or it will extend "celebrating Christmas" to include this kind of gift giving dates. They will ask "Would you put up the Christmas tree in July, reasoning that it is not celebrating Christmas since you are doing it in July instead of December" to put a stop to off-day gift giving.