This not a full theory just some observations combined with some thought.

I believe the tree doted tattoo is a red hering and based on a misinterpretation on janes part.

Watch the scene closely from the time when they start rushing towards the PI till jane gets there.

We get to see the tracking including the Sac PD we even see a police car seemlingly rushing towards the scene and we see the cop arriving in his car.

All this is creating the illusion that we know what actually happend. We are led to believe we tracked the cop from the call until he reaches the scene. We see the tracking, we see his car and we see him arrive. In reality we don't know anything about the cop before he arrive at the scene. We don't know whether the tracking signal showed anything real, neither do we know who sat in that copcar and where it was heading. But our brain combines these three possibly unrelated images into one coherent story. Seeing Cho Rigsby and Van Pelt relax as soon as the cop arrives is subconciously pushing us further toward trusting him. We trust who our friends trust.

When he arrives he comes in holding the gun in his right hand (the one with the watch) and he then uses his left trying to stop the bleeding. Also notice how the breathing of the PI gets more hectic as soon as he arrives.

When Jane arrives the scene looks almost the same as we have left it, this leads us to believe nothing relevant has happend in the meantime.
But there are a few changes. The cop has switched sides now using his right hand to stop the bleeding. Also if u look closely the pool of blood arround her head apears to be much larger than before.
When Jane leans forward to ask the PI: "who did this to you?" we get to see the policemans left arm. Just above his wrist we can see a red spot, that seems to have the form of a red triangle.
Notice that the PI only says: "red tattoo on his left arm."

Jane than sees the 3 dots on her shoulder and asumes she wanted to disclose the nature and position of the tattoo, when infact it could just aswell be that she only wanted to point at her own left arm.

We have to be skeptical about following Janes assumptions:

"Red John bugged the office!"
"Red John killed the PI!"
"Red John has a three dotted red tattoo on his left shoulder"

These three are not facts but Jane beliefs or assumptions and there could all three be false.