I think this is the best way of looking at it. Otherwise, you start with a specific style of music–generally swing or hot jazz or something else that’s well known–identify that as jazz, and then you have to explain why A Love Supreme is also jazz.
To me Nick Mason’s drumming has a certain amount of jazz feel. I think that contributes a lot to that hazy, dreamy quality in Dark Side of the Moon.
IMHO, jazz in itself is hard to define there are sub-genres, crossovers, quasi-jazz, etc.
I like** Wordman’s **modernist definition and experimenting with breaking the classical rules. We musicians have a saying that goes… “If you make a mistake once… it’s a mistake, but if you do it every freakin’ time …that’s jazz man!”
I don’t play alot of jazz myself but I jam with people who do and I always find it entertaining, evigorating and educational. The sessions do quite often go like this.
“Hey man, you can’t do that, it doesn’t fit!”
“Sure, it does… Follow me!”
I took a college course once in which the instructor said that the two defining features of jazz are syncopation and improvisation, playing near and around the melody line instead of playing the melody as written.
That’s three, isn’t it? I’ll come in again.
I think a similar saying is attributed to Miles Davis. Lawyers for his estate will contact you shortly. 
You forgot an almost fanatical devotion.
I’m hardly an expert, but I think some jazz (bop, especially) actually throws away the melody altogether and retains the underlying harmonic structure as a platform for improvisation.
Depends on what you’re talking about exactly, but most jazz does that. There’s kind of a little part to establish the melody, and then the soloists go off on their own, with some keeping more of the melody than others. Typically, though, you keep the chord changes and improvise your own melody over it, occasionally hinting at motifs of the original melody to keep it glued together. Or not. Really, anything goes. My jazz teacher taught me one way of approaching a solo is to think of a note, and then not play it.
Which is why the “suddenly bursting into a solo” (to paraphrase the OP) is usually an illusion. Each musician will likely start his solo at the beginning of the song’s chord sequence and finish it at the end of the sequence. Everybody who’s not soloing is still aware of what “bar” they’re on (what spot in the chord sequence). I would guess they already know who’s taking a solo and in what order also. This is the traditional way, at least.
Improvisation on a melody line is a frequent jazz hallmark, but it’s not a defining element of jazz. There are plenty of jazz standards which are played the same every time.
Even if a piece gives a soloist a chance to improvise, is it really improvisation if that soloist improvises the same thing every time because they’re playing that song three times a night for month’s on end in some club?
Not to mention classical jazz pieces like Gershwin’s Rhapsody in Blue or An American in Paris. There’s no one soloing an improvised section in those orchestral pieces.
Yeah, I would agree, which is why I say it’s not necessarily a defining characteristic. But when you get jazz musicians together to play a standard, there is usually a good amount of improvising going on. Then again, I would say the same of rock music. When I played in a couple of rock bands, we never played the same piece exactly the same way twice. There may be thematic elements to an instrumental solo that were repeated from one show to the next, but they were tied together with truly improvisational bits. I think, as a musician (or in my days as a sometimes musician), it would be the death of me if I played exactly the same thing every night.
OTOH, I never heard Paul Desmond play the sax solo in “Take Five” the same way twice. 
The question is sort of like asking “What is pop music?” You can probably get pretty close, but there will be enough exceptions that a hard definition is tough to pin down. In a lot of ways it’s easier to just start listening to a lot of different styles of Jazz, and sort of start to pick up on the commonalities. A lot of what is being posted here covers most jazz, but not all. Tito Puente, Louis Armstrong, Ella Fitzgerald, Glen Miller, (sigh) Kenny G, Jelly Roll Morton, Miles David, Charles Mingus, Big Bad Voodoo Daddy and Yesterday’s New Quintet. all perform various types of Jazz. Some of them you might recognize as sort of similar, some of them are hard to understand how they are the same genre. Some have singers, some are all instrumental. One of them makes their music entirely out of samples. Some have strong melodies you can dance too, some are almost atonal and arythmic. Some are based around horns, some not. Some are small groups who improvise together, some play standardized songs. Some made their music a long time ago, some are still making new music today. But there is undeniably something similar about all of them. They do, sort of, when taken as a group, make sense as a genre, even if you wouldn’t see many similarities if you just picked out, Kenny G and Big Bad Voodoo Daddy and looked at them as a pair.
That genre is jazz.
WAY back in 2005 there was a series of threads called The Essential Music Library. We did a Jazz one. Take a look as some of the suggestions. You can find a ton of stuff on YouTube now.