I bet this has never been discovered yet on this board.
Because my 13 year old site is going through such issues since October 2010, I've been going through my log with a fine-tooth comb. One thing I've uncovered is that my expected top keywords which do not show up in the regular SERPs are ranking as they should in Google custom search.
This has been like this for a couple of months. I keep expecting it to "work itself out", but it doesn't.
Any insight as to why Custom would rank my site normally?
And this isn't just one specific Custom account. It's any account that is hosting Google's custom search.
The simplistic answer would be that custom search involves fewer competitors. However, your report also sounds like some filters may not be applied to custom search, only full organic. That idea is worth some research.