Puritan Board Junior
The question is not what Luke might have recorded but what he recorded.
For sure. But even knowing the actual words in themselves still leaves the matter of rightly determining their proper significance, in context.
The issue is where the adverbial idea "along with his household" is modifying rejoiced or believed.
Again, I agree. But as I've also been saying, I think most translations get it right in their determination that, per the given context of all in the household having heard, it almost certainly pertains to "believed." The other option creates all kinds of sticky issues, as I argue in the OP.