What type of cookies does the author Rose Levy talk about in Christmas cookies book?

What Americans call cookies, other people (e.g. Britain, Australia, India) call biscuits. Christmas cookies are cookies in the American sense. The cookbook is American; if it were British I assume it'd be called Christmas Biscuits.

"Small flat or slightly raised cake" is a bit of a bad dictionary definition; though they are small and flat, cookies can be soft, chewy, or crunchy, and the texture is not really like a cake. I don't think any American would think to describe cookies as "small cakes". The commonality is that they're something you can pick up and hold in your hand without it falling apart, they're flat (of course, some are thicker than others), and you can eat them in not too many bites (though bakeries do often have pretty big cookies).

I think you let the American vs. British thing confuse you too much. The title says "cookies." In the US, this means what I discussed above. In the UK, apparently it's a plain bun in Scotland, but this obviously is not a book about plain buns. So it's clearly American cookies.

If you do a Google image search for cookies everything you see should be American cookies - and unsurprisingly, most of them are chocolate chip. Similarly, searching for christmas cookies should get you American Christmas cookies, though they're mostly sugar cookies with Christmas decorations, not the more interesting things that are in that cookbook. (I'm pretty sure this should still be true in India, because "cookie" doesn't mean much else there, and most of the internet is American English.) If you search for "biscuits", what you get will depend on where you are - in the US, I see American biscuits (like these), while you in India might see things like the cashew biscuits you mentioned.