The bank discount yield is the convention used to quote T-bill rates. It’s a stupid convention, but it’s entrenched, so there’s nothing you nor I can do about it.
The annualized yield is the real (annual) yield that you’ll earn when you buy a T-bill and hold it to maturity. If you want to compare a T-bill investment to another type of investment, you need a common yield measure; that’s what the annualized yield is.
But, about the question number 2: “another type of investment” - you mean it’s not the money market instruments, right? Because, I thought if we wanted to compare T-Bills with money market instruments, we should use r(MM) ?