Hi KoiGuy,
The JT# 3 you saw on the chuck is the interface of the arbor to the Jacobs chuck. They are two pieces.The picture you showed is the arbor which inserts into the spindle of the drill press. This is known as the Morse taper. If there is a number size on it, it is normally located on one of the flat areas at the top {narrow end} of the taper. It can be hard to see because of dirt and age. They are not always marked. From the picture it looks to be a MT2 or MT3. My Craftsman drill press has a MT3.
The Morse taper is a very robust design. There are a few reasons for it to work loose. One is dirt and fine saw dust was caught between the arbor and socket when the chuck was inserted. Also as time goes by, fine dust particles build up inside the socket and make it difficult for the chuck to seat. Once the chuck has come loose while the drill press is running a couple times, the arbor gets dinged up a bit.
There are a few remedies for this problem. First, I would be sure there are no raised dings on the arbor. Indents are not so much a problem but bumps will prevent the arbor from fully seating. Place a piece of sandpaper in your hand. Enough to cover the palm of your hand. Lay the arbor in your hand and close it to make a fist. Rotate the arbor a few times. Wipe the arbor off. Any raised areas will be noticeable because they will be shiny with dark areas encircling them. File these down and repeat.
The next step is to clean the socket. The taper mate works well. You can also use some solvent and a rag but it is difficult to clean without the proper tool. In machine shops they usually have a Mose taper reamer. But these are pricey for us home users. They run about $70 to $100 dollars. With the drill press unplugged slide your finger around inside the socket and check for raised places and using a round file remove them. Do not be aggressive filing and sanding in the socket. If you overdo it in the socket, you will have to replace the spindle.
From the looks of the picture the lower seat area on the arbor is worn. What happens as you drill, the arbor will wiggle around slightly and the chuck will loosen. If this is the case, nothing you can do will stop the loosing and a new arbor is needed.
Hope this helps.
Gary S