ok I see where you are getting confused...
we have/you want some major modes of operation (full disclosure, I, via my shop, am or have been a certified factory installer for hitches AND controllers - all the 'bigs', this redarc is my first rodeo for it)
1) nothing hooked up. you drive, you stab the brakes, brake lights come on and most modern controllers light nothing, display nothing, just lumps of under dash mounted, knee hitting plastic
2) plug into a trailer and oliver sudden, modern controllers show some sort of lighting. mebbe a red dot, mebbe a green LED, mebbe a display color. Its cuz they send a minute amount of current out the brake wire (blue) and when you plug in at the bumper and nothing else, a circuit to ground is completed thru the magnets in the brakes themselves to ground
3) hooked up to the trailer you are choogling down the road and hit the brakes like a normal person. the car stop lights come on, direct wires or gizmos (in the case of the h/l) turn on the trailer lights and modern not $20 chinese controllers show some display action...a light increasing in intensity, a number that climbs from 0 to ??, a strip of LEDs that progressively light more and more - as the braking action increases and/or endures
and something most people - the casual civilian tow-ers - never do:
4) activate the trailer brakes and ONLY the trailer brakes - manually. slide a bar, push a button (in the case of the redarc) or as the truckers used to say "grab the charlie bar" (I dont know who charlie is and what he has to do with this but its what truckers told me when I learnt how to 'truck' and the name stuck)
In this case the controller activates the trailer brakes ONLY, according to some schedule of gain (intensity) and time (duration). It helps when you get trailer sway, or descending a long hill and just wanna knock a few MPH off.
BUT, the brakes are on and while I have not checked FMVSS - and I suspect it is the same - state law (YMMV) requires the brake lights to come on at the same time.
This is not as hard as it sounds and if you tear apart your controller (not recommended) you find that yes, throwing the MANUAL activation method does indeed route 12v OUT the brake light sense wire. If this wire is spliced into the cold side of BOO, to the car, its like someones foot touched the brake pedal. In an older direct wired car, the brake controller must source enuf current to operate ALL the brake lights and in the days of the ye olde x157 light bulb - this can be many amps, or in modern cars with bulb sense, its mA of current - like in the 50mA range.
so, no idea why ANYONE would recommend a diode EXCEPT, on older cars if you have the 4 ways on, it pulses the brakes since the 4 ways are TECHNICALLY using the brake lights, depending on where stuff is hooked up the flasher output will active the controller. In this case, if the wiring is in the line BOO->brake sense-> 4way input, you put the diode in the line AFTER brake light sense with the anode facing the cold boo/brake light sense and in this case the diode has to be able to handle ALL brake light current - trickier in an older car. I can draw up the schematic - but this does not matter in the h/l. If I am reading the SCM right - and I like to think I am - toyota has already handled this in the wiring.
hence my comment, I dont think the diode should be used AT ALL, or at least not how the dealer said
*dont know about your dealer, but I installed my hitch and wiring - simply cuz I can - but also because they tend to be fairly clueless. in fact, toyotas regional customer facing support also tends to be fairly light duty - no one I am able to get to via phone or email understands the wiring enuf to give me a straight/good answer on reverse light tap - for those of us who bought h/l's to tow heavy surge brake boat things with surge discs, we use the reverse lights to lock out the brakes cuz ya cannot back surge discs against any incline - but I digress.
so based on what you said, you need to do this test: hook up to a trailer, manually active the trailer brakes - push button/slide bar/charlie bar and have a confederate tell you if the brake lights came on. You want that answer to be: yes