Can a 2-color logo be red, black and white? I’m thinking that maybe white doesn’t count because it’s the background color?
Does a 1-color logo need to be literally black with a transparent background?
Yes, you can have a 2 colors logo with red, black and white.
If you speak of standard printing (paper, stock), white is the absence of color.
If you were asked for a 2 colors logo, it’s probably to be able to print it in 2 colors Pantone (savings on inks) and because the person doesn’t want a rainbow of colors for his/her logo.
But if you’re creating a logo specifically for something that will be used on vinyl or t-shirts for example, then white might count as a third color.
But in general, 2 colors means 2 colors and usually we don’t count white as a color for a logo, especially if that white can be the background.
Does a 1-color logo need to be literally black with a transparent
It can be any color (including black) with a transparent or white background.
In case you are wondering, you can also use different density/tint from that color (gradients or gray, for example) but know that a good logo should also be created in a way to be usable with that one color at 100% of its density, ideally.
The logo Dane Arts below uses 1 color but gives the illusion of using many colors because of the lighter butterflies but it’s still a 1 color logo technically. And as you can see on the right, the same logo still look nice at 100% of density but lost a little bit of its clarity/details.
Even if you don’t plan to use the logo for special printing projects it’s important for simple every day tasks in an office to have a logo that can print well and has a good contrast no matter what; for example, photocopies don’t always reproduce the contrast properly. A black at 70% (gray) with other elements at 100% (real black) close to it might just end up looking all 100% black, and the lighter parts of the logo that very light gray might become white once photocopied! So you need to plan your logo for that kind of situation too.
Lot of designers provide to their clients 2-3 versions of the same logo; one with the color(s), one in grayscale and one in “pure black”, etc.
You can also see the 2 colors version on the top part and 1 color in the middle, and the different ways to do it.