I've Canon 600D and I use the back button to focus and the shutter to meter, I've been using spot metering for over 2 years for now and I understand what's the difference between it and evaluative, I used to use evaluative before that.
I recently got Sigma 14mm f2.8 lens, it's discontinued now but I read good reviews about it, however the copy that I got seems to have focusing problems.
I shot 4 test shots for each metering mode on a tripod in this order: eval, spot, partial, centered
I was on Aperture priority mode all the time. All shots are at 2.8. As you can see the center weighted mode is the sharpest one, my camera doesn't behave like this with other lenses and this lens has focus problem because when I focus and use the back LCD to manually override the focus the image comes sharper.
Now the question is how by changing the metering mode on the camera the focus (the photo sharpness) change?