In the monograph on gasteromycetes by Kew
Phallus hadrianii is known as the Dune Stinkhorn, because this species is supposedly confined to sand dunes in this country.
"Habitat: restricted to sandy soil, mainly sand dunes.
Distribution: locally common in Britain in suitable dune systems. More common in southern Europe where it can occur in other habitats."
The map shows a purely coastal distribution for this species for the records they had at the time the book was published.
Other than the peridium discolouring "pinkish mauve" on exposure in
P. hadriani, the key refers to its pileal disk having a crenulated margin, while
P. impudicus has a pileal disk with a non-crenulated margin.
So I guess the question is, was the peridium genuinely pink (rather than just a bit discoloured) and did the habitat mimic a coastal dune system?
I can see why John might think
P. hadriani for the first specimen, although the habitat would tend to suggest this is unlikely. I think the peridium on David's specimen is just discoloured, but not pink, so this is typical
P. impudicus.
Ken