We can even put that ideal to rest: SH brought Moghedien to a room in which Moridin was.

They are disctinct.

SH is the result of attempts by the DO to infuse Shadowspawn with his own direct will (personally, I'm rather convinced there's no new breed of DH... it's a ridiculous notion when unlike Rand we know this is impossible as Aginor was dead then, and RJ told us his genetic knowledge is totally useless nowadays as he doesn't have the technology to do anything.. if Shai'tan could create viable breeds of Shadowspawn, he would not have needed men like Aginor in the first place, and it's not only DH (fairly minor players in the Shadow's stable) he would have created new breeds of but all the Shadowspawn would have gotten more powerful... just imagine how lethal and far more useful than DH Fades that can't be killed short of balefire would be (and it's probaly the case with SH, incidentally) So I think the DH sent to Rhuidean were not any "new breed" but unique cases of DH invested directly by Shai'tan like Shaidar Haran now is (they were not sent by Rahvin either IMO - contrary to Lanfear's lie to Rand, Rahvin was part of a plan to capture Rand with her help - Lanfear was simply betraying Rhavin by pretending he sent the DH when the understanding between the four was that Sammael would be the bait - the DH, IMO, were sent by SG because Shai'tan was afraid of Rand learning too much from Asmodean and made an attempt to kill him, which failed). That is why they could not be killed short of balefire - Shai'tan was in them as the True Power and by his will and his grasp on their souls he kept "respawning", until balefire killed them before they died). We probably didn't see more "possessed" Shadospawn because they take too much out of Shai'tan, and Rand got lucky and discovered the way to stop those the first time....

It seems to me Shai'tan's direct power over the Myrddraal has kept growing through the series (their connection to Shai'tan is greater, maybe - and it might be just returning to what it used to be when the Bore wasn't sealed off). He's just invested massively more (all he had available, probably) into a single Myrddraal that would be a vessel for him in Creation after his early experiments worked. It's very telling that Shai'tan chose a creature naturally spawned by others creatures for which the TP was originally used in their making over natural beings in the Pattern, like humans. The Myrddraal are the closest thing to a real "Shadow breed".

The most telling thing about Moridin and Shaidar Haran is that Moridin is the only Chosen SH has so far left alone, and the only one who seems to work in tandem with SH. These two probably have much off-screen interaction (unless Moridin has a mental connection to Shai'tan as Nae'blis we don't know about, but that's doubftul), but so far we only see them work together but apart in complementary missions: Moridin spots Sammael doing something possibly dubious, SH follows him around to get to the bottom of it, and in the end Moridin is the one who appears to save Rand from Sammael (he might have be summonsed to go to SL by Shaidar Haran). Graendal might refuse to obey a summons to Moridin by Moghedien, so SH is around to implement a plan B to force her to come. He brought the mindtrapped Moghedien to Moridin as well. He punished Mesaana for refusing to even acknowledge Moridin as Nae'blis and ignoring his summons/orders. Moridin and SH worked together again about Semirhage - Moridin ordering Mesaana/Demandred to back off, SH finishing the job.