I’d go with the second one to fix the problem, a lot of producers advice this one also. Even if you have a vocal booth, it helps with vocal articulation. Your more used to hearing yourself sing without using headphones, so this is probably why you have a hard time being sensitive to vocal pitch. Even some singers will press one finger against the ear to help, I’m sure you have noticed this at some point. In this case it would be the headphones of course.

I haven’t even heard of the first one. Seems like a lot of work, I mean how would you get one monitor out of phrase with the other? Second, having the monitors on while recording the vocal track is quite risky. You really don’t want any sound but the singing when recording the vocal track. I know some producers will even strictly make you remove stuff like jewelry and shoes, cause it produces too much sound. Noise Gate can limit this to some point but you want the vocals to sound as natural as possible, even compressor is sometimes a delicate application.

When getting ready to record you should use the monitors to get the settings right. If you get the monitor mix down the headphone mix will sound great too. It doesn’t work the other way around. Since with vocals you can’t really do both singing and mixing at the same time. Here’s an idea. My situation may be a little different but you should be able to figure how to work it in your own situation. Now here’s what I do, I record a vocal track straight into the recorder with no effects or EQ adjustments. Then I go back and use my effects send on the vocal track and then mix it with the music. Save the settings, then go back and do the vocal track with the effects and EQ adjustments. This saves a lot of time of going back and forth singing a little, then mixing, then singing a little, ahh you get the point.