That an executed man's grave has a headstone is not a sign of "confusion" in the slightest. It was Imperial War Graves Commission policy that every serving soldier with a known grave should have a standard headstone, and there was no exception even for those convicted and executed. The view was that it was service in itself that "earned" the headstone, and being executed for any crime did not negate that. This is similar to the stipulation that a Victoria Cross could never be rescinded, under any circumstances. The standard illustration would be that a convicted murderer could theoretically wear his VC on the gallows.
I assume you are writing about a soldier who died in this way and wanting to be realistic about how the family found. This is difficult to know as he things were supposed to happen, how they were planned and whether it went either way is difficult to tell. In general a family would have been informed of a bereavement by telegram, and these could be incorrect or inaccurate. Some families asked for more information from the army, others got news from friends in the same unit (most Pals would have friends, even family, posted with them) and, as mentioned, others might hear through despatches or the Gazette.
The best thing to do would be find a couple of accounts about executed soldiers and write something along those lines. At least if someone queries it you have some basis for it.
There seems to be quite a bit online about this situation, though not exactly what you're after. For example, there's an article on http://www.centenarynews.com regarding the admittedly unusual execution of an officer for cowardice which says, 'Eric Poole was executed at Poperinge, near Ypres, in Belgium. He was 31. Original records state only that he 'died of wounds'.
This individual has a grave and headstone, which further indicates the confusion there was over this type of 'crime'.
There should be records of Courts Martial stored somewhere and these would indicate actual sentences passed. I doubt a Death Certificate would be entirely truthful as it would have been available to the public by application. Listing 'execution' as cause of death wouldn't have been good for the country's morale at that time.
You might have better luck joining one of the WW1 forums and searching and/or asking there.
Don't know, but if you send off for his death certificate it might say how he died. If it is not on it then it would probably mean the family were not informed.
I think that maybe if it was a court marshal the family might know as it would probably be made public. If he was shot for cowardice, this may have happened 'in the field' situation in which his death was just as a result of injuries.
All maybe's and don't really knows, sorry about that.
Ps anyone can send off for someone's death in the UK. It was the only way I could find out how my boyfriend died, as his relations were not forthcoming.