...but a processed black is still being found in InDesign- which I am unable to find.
It's not clear what you mean by "being found in InDesign." You say you're unable to find it, so who is finding it?
If you mean that you can't get rid of InDesign's [Black] swatch, it's true; you can't. But as long as it isn't applied to any objects in your layout, it's harmless to your 2-spot-color job. You could have 100 swatches in the Swatches panel of a 2-color job. It doesn't matter.
The printer's marks are printed in the [Registration] swatch, which is just a special black that prints on every plate.
but a processed black is still being found in InDesign- which I am unable to find.
You can use Preflight to find the black object. So define a rule that allows a max of 2 spot colors and no CMYK colors like this:
Then turn on Preflight with the 2-color profile and you will get a list of problems. Here I've double clicked the problem and the black rectangle with a 0% tint gets selected.
Also if you are printing separations out of InDesign the print dialog always shows the 4 CMYK plates even if they do not contain content. It's up to you to turn off the CMYK plates if you don't want them to output as blank separations. If you export a spot color PDF the same will happen in Acrobat. Registration marks are always on all of the CMYK and Spot plates, whether the colors are used on the page or not.
Automatic marks produced during print or export are always in Registration, which is 100% of CMY and K, plus any spots. If you want two-color marks, make them manually.