I purchased Minecraft in April, but the new download only gives me Demo

BACKGROUND:

I bought an earlier version of Mine Craft on April 23 of this year. 
I think it was version 1.5.2. 
My child's been using it off-line only.  No mods. Very happy with it.
 Two days ago, I tried to upgrade to version 1.6.2.



THE NEW LAUNCHER (1.1.2)

1. I downloaded new launcher first.

2. When I dragged the Mine Craft icon onto the Applications folder icon (in the Disk Image window), it asked me if I wanted to replace the current copy.  I said yes.

3. It warned me that it cannot do so.

4. Assuming the old version was to blame, I manually deleted the old Mine Craft file in my Applications folder, and then dragged the new one in from the disk image window. It seemed to have worked.

5. I double clicked on the icon and the new Launcher 1.1.2 opened (after showing some command-line code for a while first).

8. But where it should say "Play", it says "Play Demo" instead.



VERSION 1.6.2

1. Went to Minecraft.net  and clicked: "Already bought the game? Download it here."

2. It took me to the download page, where I clicked on: "Minecraft.dmg"

3. But no matter how many times I tried, it keeps downloading the demo only, not the actual game.

What more can I do get MinceCraft 1.6.2 working on my machine?