One difference would be the angle you body is in, relative to the pull of gravity.
This can be compared to incline/neutral/decline bench presses/tricep extensions, where similar changes of position are used.
By changing the way your body is positioned, you will have to use different stabilising muscles (mostly the upper/lower chest in varying degrees).
Another reason for changing weight placement is that the moment arms change:
- When hanging the weight from a dip belt, it's directly under your body, your arms etc.
You won't have to fight any swinging motions, as there are no moment arms.
- When hanging the weight from neck region (like in your first picture), you'll have a considerable moment arm, that constantly tries to pull you over. To avoid toppling off the dip station, you'll have to actively work against that moment arm and stabilize your position. This will add additional stress on your stabilizing muscles.
- An extreme example: When putting on weighted cuffs on your ankles and extending your feet in front of you, this becomes much clearer. With this variation, you'd work your abs much more than normal, in addition to your arms, chest etc.
Other positions might influence body position, moment arms and stabilizing needs differently, using different muscles to varying degrees.
Almost all of these will put a different stress on the body than the 'basic dip', so an injury risk is definitely there, as the muscles haven't been used in that way before.
Also consider that the longer the moment arm between the weight and your shoulders, the more strain is put on your shoulders. So most variations put additional stress on the shoulders, which might heighten the injury risk if you lose form or overdo it.