Not only did Adam want to prove himself to Cole through a battle, but he also wanted to go all out for the first time. When he fought with Chase he had to hold himself back as he didn't want to injure him and he also didn't want to show his abilities to Elizabeth and Jaspal, while against the triad members, he didn't have the chance to as their battle was short and his main focus was to quickly kill them.
In a life or death battle that was serious and between enemies, Adam didn't have the luxury of playing around and just in case they could call for reinforcements or had a secret ability, went straight for the kill.
However, since he completely trusted Cole, he could unleash more of his full power and it would allow him to find out just how capable he was and with his current capability, what level of cultivators he could deal with.