Love your brewing space!
Steps 3 and 4 in your schematic is showing both BK and MT connected to the pump input.
My apologies about steps 3 and 4 in the process diagram, it's a quick reference sheet I use while brewing. For clarify, there is only one green silicone tube. In step 3 when the green tube is connected to the bottom valve of the MT it's recirculating the mash, and if it's connected to BK for an infusion, depending on the desired mashing profile. In step 4 three operations happen, 1. connected to the MT transfers the wort to the BK, 2. connected to BK allows RIMS to help the BK bring the wort to boil, 3. Close the BK outlet valve and open RO valve to flush wort out of RIMS/pump system and into the BK to boil. Now step 5 will make more sense, there are 4 processes that occur with the exact same tubing setup. The little circles at the ends of the colored tubes represent valves and blacken in means closed. Sorry, this whole paragraph seems convoluted.
More simply put, only one end of the blue tube and only one end of the green tube are ever moved during all the brewday processes. The green tube is the inlet to the RIMS/pump system and the blue tube is the outlet. The RIMS/pump system includes the RIMS heating, wireless probe temperature monitoring, a valve flow controller and a flowmeter for gal/min measurements. When I realized the simplicity of thinking of the RIMS/pump system as a stand alone device, I was seriously excited about it.
Quick process outline I use - 19 gallons RO pump to BK, add salts and heat with BK & RIMS, 19 gallons strike water under let to MT, don't stir, recirculate & heat with RIMS, 19 gallons wort pump to BK, add dripping from MT by hand up till 30 minutes into boil, RIMS/pump system helps BK to boil stage, RIMS/pump system RO flushed into BK, boil 75 minutes (without RIMS/pump) & boil additions, sanitize CFC last 15 minutes of boil, cool to 160F for 20 minute hop whirlpool, continue cooling to < 80F, and settle wort over night. I prefer the simple full volume approach which is not as efficient but fits my philosophy of best the possible beer for the least effort.
Definitely want to add a RIMS tube to the mix for better mash temp control.
I would definitely recommend a RIMS for quicker heating throughout the brew day, and for me it's the ability to temperature infuse from 145F to 160F to get the highest fermentability. Checked out brewhardware.com for RIMS stuff and also when you want to electrify your BK. The site proprietor, Bobby_M a frequent HBT contributor, has excellent videos on the electrical/hardware applications and installations.
I am still working out my equipment profile and trying to figure out my volume and temperature issues
Using BeerSmith software comes with a somewhat steep learning curve, but after fine tuning for your setup it's well worth the effort. I document everything in BeerSmith for the convenience of troubleshooting, recipe development and all the brewing hardware and procedural updates.
BTW, I think the best taken away from the above procedures is to under let the mash, don't stir and recirculate. It simplifies mash in and by not suspending the fine particular matter in the grist a stuck mash is highly unlikely to happen. Efficiency is not effected, though my mash profile does take 90 minutes because of the low temperature, 145F, beta amylase start.
So you know, I share what I have learned as a form of paying it forward. I'm always looking for better ideas. A tinker always changes the process. Enjoy!