According to David Chapman, modern Zen was basically invented by D.T. Suzuki, and doesn't have much to do with original Zen or even with Buddhism in general. Is this true? Is modern Zen closer to other Western Buddhist traditions than to original Zen?
Someone else will answer this better than me, but I'll get the ball rolling.
Chan is the form of Buddhism that came to dominate Buddhist life in China up to the time of the communist revolution. It was a lot of meditation, mixed in with a lot of Pure Land Buddhism, scholarship and the whole gammut of Buddhist practices, including prostrations, repentance, everything.
Chan was then transmitted to Japan via Dogen around the time of the Kamakura era. At this point, Zen is radically simplified, loses the Pure Land component. Dogen is a medieval Japanese monk-- he's a nice guy, but no one is going to mistake him for a modern Oxford or Tokyo University intellectual.
Also at this time, when Japanese nobles needed to send their kids to school, they would send them up the road to the Buddhist monk, who knew how to read. While they were there, the parents figure the should also learn about calligraphy, tea ceremony and things that upper class Japanese noble kids needed to learn. This created the confusion about Zen's connection to these Japanese cultural arts. (And this confusion was later exacerbated by apologists for Zen, who wanted to appeal to skeptical, but nationalist countrymen in the revival or preservation of Buddhism)
Next big event is the Meiji reformations, where the Japanese imperial government, in a policy hostile to Buddhism, initiated reforms allowing (demanding?) that monks marry, eat meat and so on. From this we get the idea that Zen has nothing to do with strict morals, renunciation, vegetarianism, celibacy, and so on.
The last big event is the educated class in Japan learning pretty much everything that the Europeans knew and being heavily influenced by essentially European ideas. (among them ideas of the age of enlightenment and the protestant reformation) Not bad mind you, as a 3rd party, I could care less if the Indian ideas imported into Japan were being diluted of their Japaneseness by mixing with European ideas.
And the person to bring that "pre-westernized" Zen to the US was DT Suzuki. Before anyone had a chance to start mixing up western progressive ideas with Chan (as filtered through Japan), it had already been done. DT Suzuki Zen removed anything that looked like religion from Zen.
"Original Zen", it is said, was started as an act by Mahākāśyapa (one of the original 5 disciples of Gautama Shakyamuni. He gave a knowing look to Gautama when he held up a flower.
The genealogy of Zen traces from that moment, to an Indian monk (Bodhidharma) who's master told him to go to China. That's the mythology anyway. For the full scholarly picture read John R. McRae (d. 2011).
From this single "transmission" down through the present, the message- stripped of all pomp and circumstance (orthodoxy) has not changed one iota (on itoa):
Practice* until you "see through" to your Original Nature (have a kensho experience - preparing for this using the precepts and other "expedient means" and "skillful devices" (koan study) to season/ripen your consciousness) then practice continually for the rest of endless time (we're always just beginners) to integrate that seeing through into all activities.
Side note: The kensho (enlightenment experience) is just the first part of training and is not even considered anything special. That is why an affectation in Zen for enlightenment is "Ordinary Unexcelled Enlightenment". It's what you use to "carry water and chop wood" (daily activities lived in the present moment).
*Varies based on time, geography, school (Rinzai, Soto, other), and teacher.
EDIT: Andrei pointed out that I only answered your first question. As to your second question- I know of no Buddhist traditions I consider originating in the West. All are flavors (not schools in the traditional sense but more like genealogical lines from prominent masters) of Eastern traditions. See here for more on Buddhism in the West.