Pictured are two stainless steel 25mm connector rings. The ring on the left is one where the wire has been bent after attaching it to a large D-ring.
As you can see the mini ring is thin enough to get caught in the gap, and all it takes is a dog shaking off or bouncing around for a bit for it to slide off.
For the smaller ring to stay securely on, it needs to look like the photo below.
The better, more secure option (if you prefer a ring as a connector) is to connect the larger ring directly to the tag. The reason for this is that the material between the hole of the tag and the edge is much thicker than the mini ring and - unless there is a MUCH bigger gap in the wires - it is very unlikely that your tag will slip off.
So why do some people still request the mini rings?
With the mini connector, the tag hangs facing forward. For anyone who wants the tag to hang forward but also wants a more secure option, I recommend using an s-hook with a mini ring or a 19mm ring.
Without the mini ring, the tag will typically face to the side.