You are correct. The market price of bonds will normally fluctuate up or down so that YTM is in line with prevailing interest rates. Thus the bond will trade at a premium or discount to its face value. (YTM = effective yield assuming the bond is held to maturity.)

The reason for this is that bonds with a coupon rate greater than the current interest rate becomes desirable. And they are willing to pay to get it.

No. From the standpoint of the bond issuer, a bond discount or premium is the difference between par and the proceeds from the sale of the bonds. If you have a bond with a par of $100 and issue it for $105, you have a $5 premium. Issue for $95 and you have a $5 discount.

This is my understanding of bond premiums and discounts. Please correct me if/where I'm wrong.

Usually bonds are issued at face/par value right? If interest rates increase/decrease after it is issued, the market value can decrease or increase as an adjustment. The difference between the par and the current market price is the premium or discount. Is this correct?