One says YES, and the other says NO???

Different orthodontists do braces differently. Many orthos extract teeth on virtually all patients to make room to spread out teeth and straighten them. This works well, but isn't necessarily always necessary. Many orthodontists are now cutting back on pulling teeth. It may help with your bite/TMJ long term to avoid popping and cracking/ pain. All methods generally work. It just depends on your orthodontist and what he/she was taught. Just make sure the 2nd dentist has a good background/reputation in ortho.