I would count any trigger that fully cocks and releases the striker/hammer AND mainspring with every shot as DAO. The Kahr does not count because the striker starts partially cocked, which is why it is not able to second strike.
I would not call any short reset trigger DAO. That second shot on a Glock or LEM after a tiny reset is certainly not a "double action". There is only one action performed at that point, since the striker or hammer fully cocked after each shot.
The LDA, like the LEM, Daewoo Fast Action/Double Action + and FN's SFS are all systems that use completely cocked mainsprings with uncocked hammers. In terms of pre-cocked springs, they are arguably SA. The fact that the hammer is out of position is nifty, but somewhat meaningless. More of a nice feeling than an actual mechanical safety mechanism. FN first came up with this idea in the early '80s, before the Glock came out. It is the granddaddy of all the pre-cocked mechanisms that followed.
Beretta 92Ds, especially worn in ones, have beautiful triggers. You can duplicate that for the first shot with 92 DA/SA guns as well. The 250 may well be better.