Diner's Club is equivalent to Discover card and is at least somewhat present in European countries. I don't know official numbers but I've seen it often enough and have used it once or twice.

“Amex is virtually unknown in most of Europe” doesn’t match my experience. Most major European economies have AmEx and AmEx issues many local cards to them

Fun fact: AMEX has been involved in international payments in one way or another for more than a 100 years and the AMEX card was the first international charge card, launched the same year as the original BofA Visa Card.

Doesn't Discover have agreements with UnionPay and JCB?