More than a year too late for you, but in case it still matters, or in case someone stumbles across this thread as I did, here is the solution:
The trick is the setting called “SDI Output Always On” on page 2 of the System Setup menu. On my C200, this was set to “On (Linked to VIDEO Term.).” I discovered if you set this to “Off” instead (which wasn't at all intuitive to me), then finally the SDI output will fully obey the “OSD Output” setting over on page 6 of the Monitoring Setup menu. With “OSD Output” set to off, and “SDI Output Always On” set to off, then the SDI output will finally be a fully clean signal.
The additional gotcha that really stumped me for a while was that as long as “SDI Output Always On” was set to “On (Linked to VIDEO Term.),” I also could not get audio to pass through SDI. Only with “SDI Output Always On” set to off would audio pass through. Very strange.
But let’s say you wanted to still be able to see the onscreen displays through SDI, like if you were monitoring through an external monitor or recorder and wanted to be able to see things like exposure, ISO, and white balance settings on that external device that is connected via SDI, but then also wanted to toggle all of that completely off (including “STBY” and “REC”) so your SDI feed is completely clean. Well, you can do that too.
The way to do that would be to have “SDI Output Always On” set to off as above, and then set “OSD Output” to on. Now when you toggle the “DISP” button on the EVF, most of the onscreen displays will appear or disappear from the SDI output. I say most because “STBY” and “REC” will still be visible on the SDI output, as OP discovered. To get rid of those, you go to the “Custom Display 2” setting on page 6 of the Monitoring Setup menu, and toggle “Recording Mode” to off. This will completely turn off the “STBY” and “REC” notifications on both the EVF and SDI output, regardless of whether you have the DISP button toggled on or off. The downside there is that you will have to rely on the tally lights to tell if you’re recording (if you’re using the camera itself to record).
Perhaps with 4K being more and more prevalent, and the C200’s SDI output only capable of sending out up to 2K resolution, all of this information is too little too late. But I figured I’d post it for posterity anyway.
To this day, I still have no idea what the “SDI Output Always On” setting is even for or what its two settings are meant to accomplish. And I have no idea what the relevance of “Linked to VIDEO Term.” is, but it certainly isn’t intuitive for me.
