Courting is simply dating with a purpose. It has a serious intent without serious pressure. You date in order to see whether you are meant to be married, not because you think each other are cute and like each other (even though you may like each other as well). The relationship is based on wisdom, not feelings. Even though feelings are good and necessary, they shouldn't drive the decisions that each individual make in the relationship. A Christian courting couple are basing their decisions on what God's will is, not their own. And when they figure out they're not meant for each other, then at least they knew where they stood in the relationship because of the definition of it. These two are typically more apt to stay friends unless they got their emotions intertwined with their decisions and didn't let wisdom guide them.
I don't quite understand the whole switch from "dating" to "courting". Maybe he thinks there was no definition in the relationship or he doesn't understand courting himself. In Joshua Harris's book "Boy Meets Girl", which is devoted to defining and guiding courtship, he talks about this one incident after a conference when a 13 year old boy walks up to Joshua with his girlfriend in hand and proudly says, "We stopped dating. Now we're courting!" He didn't get it. lol He's not ready for marriage, obviously. Talk to your boyfriend and see what he thinks courting is. He may just have the wrong idea.