There is new terms like "wordpress hosting" but usually this is just marketing and a regular shared hosting plan is the same thing in fact.
Depends on the host, but yes, there's a bunch of places that claim "optimized" etc, but they don't go the extra step.
I am with a couple of hosts that truly are Wordpress Optimized with extra server features and restrictions. Some only allow you to upload plugins and themes, the core of Wordpress is in a different environment and optimized. Those I would trust more than someone just claiming optimized.
The big trends that I'm seeing now, and they've been around for a while, are backups and cloud hosted. Many hosts have moved away from the individual server and are now using some form of cloud or grid.
In the SEO world, TTFB (Time To First Byte) is something they watch for, and I've begun to see hosts claiming to have the lowest TTFB or showing their TTFB.
Years ago, hosts used to offer a sample 5MB, 50MB and 100MB file that you could download and see how the speed is. This appears to now have trickled over to the TTFB and similar requests.