You may be over thinking this.
I don't see why you want to record with omnis and directional mics at the same time. The characteristics of one type or the other type are going to be better for your particular situation, which is why units offering both have you switch between them. Mixing the two types would be something a pro would do with a multitrack setup. (So perhaps there would be a cardioid/directional up close to the instrument, for detail, and omnis further away for the warmth of the room.)
And I don't see why you think you need more than two channels. Are four or six channels really necessary for you?
If I understand correctly, what you want to do is get a recorder with built in mics and put it in the room where you are performing and hit the Record button. You can do that with the Sony PCM-M10 quite nicely, and with the money you save over a fancier recorder you can get some nice mics. The Sony D-50 is apparently discontinued, but the PCM-M10 is going strong. However, if your microphones have XLR connections, the Sony does not. But if you are just planning to use the built-in mics, that's not a consideration.
The professional recordists here may be right that the PCM-M10's omnis do not create the most richly, perfectly realistic stereo image (because they are too close together, not separated as far as your ears) but you're just recording a rehearsal to see how you sound, not making a professional CD. You'd be fine with the PCM-M10. Even better if you used it with a pair of outboard mics that you can now afford.
If you are in Europe, the Sony is way more expensive. Another similar recorder, more sensibly priced in Europe, is the Roland R-05. Or you could have a friend who's visiting the US grab a Sony for you...