Your list seems to be a good start, although I would have to argue against one Country/Western/Bluegrass category.
To be anything near properly indicative of the genre, both historically and today, you would need, IMO:
Country & Western: to include traditional country & western, texas swing, honky tonk and popular country music up to the Rhinestone Cowboy era.
Examples: Loretta Lynn, Johnny Cash, Waylon, Patsy, Hank, etc.
Modern Country: late 70’s country onwards to today - this is a pretty weak genre, although it’s a wildly popular one, with plenty of hits. This encompasses everyone from Garth Brooks and Lee Greenwood to Shania, Billy Ray and Alan Jackson. Half of it is really rock with a slight swing and a heavy accent.
Alt-Country/Americana: modern country that doesn’t fall in the the above category - this is a wide category, but a fertile one. Examples include Uncle Tupelo, Old 97’s, Wilco, The Jayhawks, etc. There’s a recent thread talking about alt-country, but I’m too lazy to search for it. It’s got plenty of examples and a good knowledge base.
Roots/Americana: I spin off the acoustic side of Alt-Country to include such artists as Gillian Welch, Lucinda Williams, Jay Ferrar’s solo acoustic work, Rhonda Vincent, Bob Cheevers, etc. A good listing of this type of artist is here.
Old Time: This is a traditional genre - to include Piedmont Blues, Appalachian Music, French Canadian, and English derivatives (based on Irish, Scottish and Northumbrian traditional music), as well as sea chanteys, whaling songs, and such. Artists such as Doc Watson would fall in this category, as well as such albums as Garcia/Grisman’s Shady Grove, which I would consider a must-have by the way.
Bluegrass: The classic high lonesome sound - bluegrass is a very specific genre, started by Bill Monroe, with a unique sound and history. Giants of the genre include Jimmy Martin, The Stanley Brothers, The Osborne Brothers, Del McCoury, Don Reno, Flatt & Scruggs. There was a significant ‘Bluegrass Revival’ that came about in the late 60’s-70’s, which also spurred significant popular interest in old-time and other folk traditions. This paved the way for Newgrass, which is a modern updating of the tradition - major lights of this style include the Peter Rowan, Seldom Scene, King Wilkie, AKUS, Sam Bush, Bela Fleck’s 'grass work, and so on.
Jamgrass: a variant of 'grass featuring a stronger emphasis on group improvisition yet still well within the 'grass world. The best known name would be Yonder Mountain String Band, but also to include Foghorn Stringband, Hot Buttered Rum, Tim O’Brien’s more recent work, and some of Grisman’s later work (although he crosses a lot of genres and is more rightfully his own genre which is called Dawg Music.)
There’s more, but at the risk of seeming overly pedantic, I’ll stop here.
And don’t forget folk and other non-country/mountain based acoustic musics…
BTW, if you want to really dig into to the 'grass world, you should start with BluegrassBox.com as it features the best of the web in terms of free, legal downloads of complete shows, a bunch of good radio streams, and a good representative catalog of the bluegrass/newgrass/jamgrass world, and a good bit of the grassier old-time music too.