Seems to me that different jacket styles are being discussed.
Here, on the right, the King wears a doublet.
But the jacket on the left cannot be a doublet, as it appears to lack the flaps.
It could be a coatee (if it has tails in the back) or a shell-jacket (if it ends at the waist all around) but without seeing the back we can't know.
What the King is wearing appear to be the Kenmore Doublet, which the firm Andersons, in their catalogues from the 1920s and 1930s, claim to have recently invented.
Here's what they say (the illustration is of a Coatee, not the Kenmore Doublet) which implies that they made the Kenmore Doublet both with stand collar and with lapels.
This illustration shows their Kenmore Doublet with open collar.
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