When multiple triggers happen at the same time, the turn player decides which player resolves their triggers first, and each player orders their own triggers. If there's only one trigger involved for each player, that basically means the turn player gets to order them.
AFAIK this is a case of simultaneous triggers, so the warrior player can choose.
#Decimator vs barkskin of mel. Tree
halcyon rivet
tawny shale
Note that any order of resolution for the Barkskin and Decimator triggers will ultimately produce the same result for the defending card chosen by the Decimator trigger.
The triggers in this example all create continuous effects upon resolution, and there are rules for how multiple continuous effects are applied to the same object.
Decimator halves base defense and embodiments of earth add +1 defense each, and these effects are applied in different stages, with the adjustment to the base value being applied first.