Is what a Diners card or not, in what sense?

There's acceptance network, there's issuer, there's owner, and there's program.

Diners Club cards issued in the US are on the MasterCard acceptance network worldwide (they have numbers that start with 5, which automatically means MC), they're issued by and owned by Citibank, and the program (partners and benefits) is Diners Club (Club Rewards). The fact that the acceptance is MC and the owner is Citi doesn't change the fact that the program is Diners Club.

The acceptance network for cards issued overseas is more complex. They have the MC logo only on the back, and those cards are supposed to be accepted in the US as MCs (tho some people have had some problems) but elsewhere in the world they're still accepted only on the Diners Club network.

(Canada is similar to the US in terms of acceptance, ownership, and issuer, but the partners and benefits may still be different.)

And, at present, Discover has nothing to do with any of this. The Discover deal is an announcement only so far, nothing has been implemented yet. (And not even details have been announced.) Our best understanding is that the Discover acceptance network will merge with the (non-MC) Diners Club acceptance network worldwide. It won't affect the MC acceptance (5xxx), but it'll change (presumably widen) the non-MC Diners Club acceptance (3xxx, or maybe they'll change to 6xxx?).

Meanwhile, issuers and ownership of Diners Club overseas is a patchwork, and the programs are all different too. In some countries it's also owned by and issued by Citi, but in other countries it's a different bank. The airline and hotel partners are different for cards issued in each country, the earning rates can be different for cards issued in each country, and the other benefits can be different for cards issued in each country (whether or not the bank is Citi).