Thanks for everyone's contributions, but it isn't any of the suggested causes.
I think I have found the answer, but it is very specific to my setup, so read on if your audio interface is a focusrite saffire pro 24. It's a great interface, but it comes with some demonic software called saffire mix control. You are supposed to be able to add reverb and other effects for the benefit of the vocalist without necessarily recording it. But it is recording it. Only when you switch the reverb off does it stop. Even quitting saffire mix control doesnt help if you havent switched the reverb off. I know that sounds crazy. Maybe it's me.
Mix Control, at least what came with the Saffire Pro 26 I used to use, does not add any FX, i.e., there are no downloadable or built-in DSPs in that line of interfaces. However, Mix Control and Logic, together, have a slightly confusing (IMO) way of doing/presenting the round-trip monitoring vs. direct monitoring settings, and I believe you're getting them mixed, or at least confused. (I no longer use my Saffire Pro 26, or I'd try to recreate how you are doing this with pictures - too lazy to hook it up and re-install the app.)
If reverb is off in Logic, it won't get recorded. However, if you have it on, but doing "direct monitoring" (in the Saffire/Logic way) then you'll hear a nice, dry sound in your headphones, but the recording will have reverb.