How many patch leads can I put through each cable ring?

There is no useful universal cable count. Cable diameter varies too much for a number such as "15 leads per ring" to mean very much: fifteen slim patch leads and fifteen substantial Cat6A leads are quite different propositions.

The 19" rings are also physically larger than the 10" ones and naturally accommodate a larger bundle.

I would treat capacity as a serviceability question rather than a packing competition.

Once adding or removing one lead becomes a wrestling match, divide the route instead of persuading another cable into an already crowded ring.