Dear Brother,
I agree with you that most definitely there is a difference in how one identifies oneself! However, to be honest I rarely if ever find myself in a situation that I start out with "Christian" as an identifier, unless its to differentiate myself as it pertains to religions or with the filling out of paperwork. Otherwise, I use something similar to what you used "child of God", "A Child of the Most High", which of course makes them as you say "pause".
I also find that I use Christian as an identifier more in writings, then in speech. Probably because, the t-shirts I wear, or even the masks I wear have Scripture verses, Crosses, or something that even a non-believer would associate with Christianity. I'm of the belief, that if I'm going to advertise for someone, i.e. company brand, I'd rather do it for the Lord. Doing so, has allowed me to have some interesting conversations over the years. People will stop me and ask where I purchased it, and this open it up to a deeper conversation of Scripture, my Lord, and where they might stand. To give you an example. One of my favorite t-shirts has a coffee cup and the word Coffee with the words for each letter as follows:
Christ
Offers
Forgiveness
For
Everyone
Everywhere
The fact that I also love a good cup of coffee in the morning, is just a plus.
Taking every opportunity to be a witness to a ever darkening world, is what I seek to do in my walk.
With the Love of Christ Jesus.
YBIC
Nick
\o/
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P.S. One thing I do notice about Catholics. Most don't use "Christian" to identify themselves. They'll say "I'm Catholic" instead. Sometimes I think they don't want to be confused with some denomination that isn't Catholic.