Note: I don’t think Russia chose to engage the US in battle.
I think the Wagner group acted at the direction of Iranian forces (probably Quds) or possibly (unlikely) by direction of Assad. Every case I’ve seen where US actually hit Syrian military forces, it was at the direction of Iranian elements.
Neither Putin or Trump want to have a direct conflict. But our military will retaliate if fired upon. No questions asked.
I am like 90% sure the clash was about a private or corporate oil interest and has nothing to do with the Russian government. The only thing linking Wagner to the real military is they are mostly veterans but most of them had a history of bad conduct ending with dishonorable discharge or even conviction. That is how they end up in such groups.