It is confusing, and maybe I've added to the confusion. I wish there was standard terminology for this.
Assume we're printing a step table, and assume the Photoshop document in Gray Gamma 2.2. Since the image is a step table, it does not need to be inverted.
If we send this image to the printer and specify Color (not B&W) the printer will make neutral tones by mixing C,M,Y, and K ink. The resulting print is what I may have called "CMYK grayscale" to distinguish it from a grayscale negative made with black ink only, which you would get if you chose B&W in the printer settings (not a good idea).
To make a colorized grayscale negative, we convert the document to Adobe RGB, then add color. The original Burkholder (1999) method was to do a Fill -> Color with the Fill operation in Color mode. The way I would do it today would be to add a Color Fill layer in Color mode. Some people use Screen mode.
If instead of a step table you were printing a real image, you would invert before colorizing.
There is another method of colorizing that is used by the "Easy Digital Negatives" (EDN) system, and I believe it explains why colorized negatives remain popular. EDN colorizes by applying a gradient map over the image (or the inverted image). The "stops" of the gradient attempt to shape the printer curve so that it's closer to linear. It's an elegant idea, but it suffers from "continuity" problems in the midtones. As tnp61 pointed out, it's not going to make much difference in a typical alt process print. But a mathematical model chokes on inversions.
I wanted to understand the history of digital negative making, and so I found a used copy of Burkholder's 1999 book on eBay. Back then, everyone was making negatives with imagesetters. But affordable inkjet printers were becoming available, and he talks about his experience using them. This is where he explains his decision to make colorized negatives, rather than CMYK grayscale negatives: colorized gives you equal density with less ink, but not as good coverage (it was Burkholder who distinguished between "physical density" and "spectral density"). By 2013 (his "Inkjet Companion" book) he had abandoned color and gone back to CMYK grayscale. A grayscale printer curve (transmission density vs. digital tone) is a thing of beauty, even on my "K1" printer.
I use your printer settings (see below) !!
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