I've written the patterns in the now-standard triplet form, but swing rhythm is more complicated than that; even more so with swing rhythm on the cymbal, which can vary dramatically depending on the player, the tempo, and the style. That's beyond the scope of what we're talking about here— just don't be too triplet-fixated in your swing interpretation.
And don't overdo it with the accenting; if you listen to the music, it's fairly subtle the way the players do these things. You have to find a musical balance: if the accent obliterates the other notes of the pattern, you'll be blowing a hole in your time feel. And the cymbal is going to be making this pattern all night, and if you're hammering the 2 and 4, well, people's ears will get tired of that after the fourth or fifth hour.
50s / bop / Blakey-like
For a long time this driving interpretation was just the way you played the cymbal. This is generally played repetitively, with a moderate-to-strong accent on 2 and 4. Listen to Art Blakey, Max Roach, Kenny Clarke, Paul Motian, Philly Joe Jones, Art Taylor, and many others.
In attempting be “loose”, it's easy to play the unaccented notes sloppily, which can drop the bottom out of groove— so, articulate the 1 and 3 even as you accent the 2 and 4. This interpretation is usually supported by a consistent, strong hihat on 2 and 4, so, again, it's not necessary to overdo the accent on the cymbal.
Quarter note pulse
In recent years I've been favoring this as my default way of playing jazz time. It communicates a strong groove to the other musicians and to the audience. I've heard it be effective even when a drummer's cymbal pattern doesn't sound very swinging by itself— as soon as the band comes in, it sounds great.
I don't know if this feeling is accurate, but it seems to me that I've most seen this approach in veteran players who were young in the 70s and 80s, when big-drum Tony Williams influence was at its peak. Younger players seem to favor the other styles, and you can really sound different if you emphasize and sustain this type of feel. That's just my impression— maybe it's BS.
Accenting the “skip” note, sometimes “ghosting” or omitting the note after the accent. Everyone does this today, but it was once very special. When I figured out that it was a thing, I thought I was the only person in the world doing it. This is especially helpful in making a two feel without sounding hokey. It also tends to make the music lay back, so you can deploy this if you need to make that happen. Doing this in an exaggerated, overly-regular, over-practiced way can easily sound stylized, so beware.
This is often done in a mixed rhythm, with a strong pull towards a dotted quarter note pulse:
Listen to drummers who do this interpretation, especially Elvin Jones, and in the 1960s and later, Pete LaRoca and Roy Haynes. In his jazz playing with Chick Corea, Steve Gadd does a form of this, with a legato, tending-towards-straight-8th swing interpretation.
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Just the facts
Some drummers with a great time feel seem to just play the notes of the cymbal rhythm without putting a lot of obvious shape on them, seemingly with roughly equal emphasis— Billy Higgins is an example of that.
Approach this with caution. Nobody talks about this as a valid way of playing the cymbal, yet there it is on recordings by one of the most beloved-and-recorded jazz drummers ever. Practice it as an experiment, and as a possible antidote to getting too locked into one thing. It can also be helpful when playing difficult tempos, when playing too much “feel” can cause you to rush. Instead of thinking of it as “playing all the notes evenly”, think of it as combining the quarter note pulse and the Elvin way... which happens to make you play all the notes pretty evenly.
Variations in rhythm, or the lack thereof, are also an important part of your cymbal concept. For me the classic example of this is Madness, played by Tony Williams— he mostly plays a strong quarter note pulse there. McCoy Tyner's Passion Dance is possibly an equivalent example in Elvin Jones's way of playing.
This mixed rhythm can be the result of an interactive approach, as we'll see in the next example, or it can result from making a deliberate counter-melody out of the cymbal pattern— that's what's happening in Madness.
This type of rhythm is often called broken, but what it really is is interactive (in Bob Moses's language, non-independent, or, dependent) with the other parts of the drum set. So, an irregular cymbal rhythm like this:
Suggests a whole-drumset approach, filled out something like this:
The one wrong thing
Accenting the 1 and 3 is the one thing you basically never do, unless you're going for a comedic, opposite-of-swinging effect:
...which is something a few kooky people do actually try to use. They'll play to excite some chemistry on stage by occasionally doing things not considered to be grooving or hip in normal playing. And doing wrong things for a moment can help lighten the mood on a bad gig. So never say never.
Drumming today is in a different place than it was when these approaches were being formed. Virtually no one just plays bebop their entire career, so having one cymbal interpretation that is just your thing is out— you have to know and play the whole history of the music. Today I may do all of these interpretations in the course of an evening's playing— and a lot of non-swing styles, as well. But the way I got there was through being focused on any one of these almost exclusively— for a period of years, even— as the way to play the cymbal. This was based on the music I was listening to; I was so into something that I thought it had to be the only way to play the drums. I was not checking techniques off of a list.
So you have to play a long game, being very into Elvin, Blakey, Philly Joe, or whoever, as the only way to play, until it's time to move on and let something else in.
Finally, today we make a big thing of our ostentatiously flowery cymbal technique, played on a giant, luxurious, Mercedes-Benz of a cymbal, but if you look at most great old drummers, often they'll just be playing the notes on a little, kind-of-ordinary cymbal, with a physical technique that is graceful but economical— and they sound great. That's a thing you can do.