Typefaces are a fairly rich area of creative evolution so there are a lot of different facets to them that can be described in a lot of ways.
The categories you give overlap each other, even within the same document. For example, a display typeface can also be serif or sans-serif. So they aren't all divided up into mutually exclusive categories; a typeface can belong to multiple categories based on its features.
- Serif/sans-serif refer to the presence or absense of serifs.
- Script typefaces are based on calligraphy (but usually not including blackletter, which is in a specific category of its own).
- Display/book refers to whether the typeface is intended for large short words like posters and logos, or lengthy body text such as in a book. Not every typeface is one or the other, though.
- Roman is often used to describe traditional/transitional serif typefaces that aren't italics.
- Egyptian has been used in a couple of different ways; sometimes to describe a "slab serif" and at other times it's even been used to describe sans-serif.
There are also lots of other terms such as
- Grotesque, grotesk and gothic have been used as alternative terms for sans-serif.
- Blackletter is a specific type of hand-lettering very popular before the printing press. Sometimes also called gothic script (not to be confused with either gothic or script).
- Italics are slanted forms of typefaces with some changes made to look more like hand-lettered text (that is a pretty basic, and inelegant, description).
and many more.
I'd probably generally group fonts into serif, sans-serif, script, and other (and possibly blackletter). Other may include stuff like this.