Names and labels cause confusion. The recent multi-colour lighting systems started with RGB lights (Red / Green / Blue) and use a 4-pin connection system. Then came the more complex Addressable RGB (ARGB, Digital RGB) system which contains in each light strip a string of Nodes, each with one LED of those three colours plus a tiny controller chip. Each node has its own digital address, so the instructions arriving on the digital signal line tell it which of its three LED's to light. This allows each Node in the string to be different at the same time, creating many more possible display types. This system uses three-pin connectors. The majority of the market now has "standardized" on these two design types and the connector designs. But because of differences in both voltage supplied and control signal systems, you cannot mix the two types in one lighting string. That's why you see warnings never to try to plug one type into the OTHER type of header.
It happens that MSI chose to use a slightly different label on the ARGB headers - they call them JRAINBOW, because one distinctive display that system can do is a moving rainbow of colours in one light device. Since you have the ARGB lighting type in your fans, that IS the correct mobo header.
"3-pin" vs "4-pin" can be confusing. A lighted fan really is TWO devices in one unit - a fan MOTOR, and LIGHTS. Each has its own separate connecting cable to go to different mobo headers. The fan MOTORS come in two main designs. The older ones are controlled by Voltage supplied and use three wires and 3 PINS for connection. The newer PWM design uses one additional wire and thus 4 PINS. So we talk of 3-pin and 4-pin FANS, meaning the MOTORS of those fans. THEN came the addition of lights in frames, and guess what? The LIGHTS come in two types, one using 3 pins (ARGB) and the other using four (plain RGB). So those SAME labels get used for LIGHTS also, BUT there is NO relationship between motor and light connections.
The cable from your case front panel's lighting controller has two connectors on it. While one has 4 pins (male) in a locking connector shroud as an OUTPUT to feed more Phanteks fans' lights, the other is a FEMALE connector with THREE holes. That is the one to plug into a JRAINBOW header on the mobo to get that control signal to the case's lighting control board.
You are right NOT to plug your fan MOTOR controller into the CPU_FAN header. That header controls fan speeds according to the temperature inside the CPU chip. BUT I presume this fan controller has attached to it the fans for CASE ventilation. For that purpose their speed control should be based on a different temp sensor on the MOTHERBOARD, and that is what the SYS_FAN headers use. So connect the fan motor control board to a SYS_FAN header, and use the CPU_FAN header only for a CPU cooling device (a fan, or part of an AIO cooling system).