We were going with SW Extra white but after seeing some pictures and comments, some people said it is TOO bright white

For a multitude of reasons, I wouldn't let comments and pictures online inform your opinion about Extra White.

Again, we want to undertones.

Paint colors don't have undertones, so you're in luck.

Would Pure white work better?

Based on what you've said, I doubt it. But I don't care for Pure White. I think there are prettier whites (personal opinion).

All paint colors are defined and described by hue, value and chroma.

The majority of white paint colors available in commercial paint brands belong to the yellow hue family, green-yellow hue family or green hue family.

The reason is because there's a whole bunch of human visual perception stuff with regard to this region of the spectrum. Long story short, we end up perceiving near neutrals like grays and whites from this spectral, hue family range as being the most neutral as in not having amounts of discernible hue.

Extra White and Pure White belong to the same hue family neighborhood. Pure White is at the end of the Yellow Hue Family, Extra White is at the beginning of the Green-Yellow hue family - again, same neighborhood on the color wheel.

When it comes to brightness - and brightness is a description of how lightness (value) looks - they're basically the same.

The biggest different between EW and PW is CHROMA.

Chroma tells you how colorful or neutral a color looks.

Pure White has more chroma. 0.44.

Which means it's not as white as Extra White which has a chroma of 0.32.

Extra White is a whiter, cleaner color than Pure White.

(remember swatches you see online can not mimic in real life colors so whatever you think you see in swatches online you should ignore and go get physical chips)