The only difference I see is where you came up and down with the needle,
therefore I don't see why there is a difference between half stitch and
You are correct that the only difference between half and tent stitch is where you come up and down. The differences are noticeable though.
Tent stitch is used primarily in needlepoint. Since needlepoint is usually done on canvas, adequate coverage is important. Also for something that may be used to be handled, say a pillow or chair cover, the more thread used the better it will last.
In cross stitch, the material is a closer weave and good coverage is not as difficult to achieve. If it's something that is going to be framed (or otherwise not handled), there is no need for the extra thread for durability.
Here's an example showing both stitches from the front and the back. The amount of thread used doing tent stitch is over double what you would use in half stitch.