Looking for dslr camera

I love DSLRs. I own several. But what you are describing with your budget will be difficult.

Cinematic typically means narrow depth of field to most people. A narrow DOF makes focus, especially on moving subjects tricky. You are right in assuming that many of the pro's would use a DSLR for these types of shots. But they have lots of experience and can typically keep a moving subject in smooth focus with a manual setup. This is very difficult for most newer fans.
You can sort of cheat. But to do so you need to understand DOF a little. Apparent DOF (or subject isolation) is based on several factors:
Sensor size - the larger the shallower the DOF.
Focal length - the larger the FL the shallower the DOF (focal length is expressed in MM)
Aperture - the smaller the aperture the shallower the DOF.
Distance to subject - the closer you are to your subject, the shallower the DOF.
Distance to background - the further away the background is (relative to your subject), the blurrier it will be.
An example in use:
Normally for portrait photos you need around a 2.8 or smaller f-number (the smaller the f-number, the larger your aperture. This is because the f-number is a ratio not a specific value). But in this example I achieved pretty good subject isolation even though I used a f-number of 4.5

This will be useful to know since your budget doesn't leave a lot of room for quality lenses. In fact most of my lenses cost more than your entire budget.

Next is the auto-focus vs manual focus issue.
If you think you can manually focus, then I would suggest a Canon or a Nikon. I am a former Canon shooter, but I am not partial to Nikon. There are skilled shooters who select both brands. These are the top two brands both in sales and in lens selection (which is all important). Canon sells a little better, and is a solid system. Nikon has a few more lenses and lower ISO noise and higher dynamic range (particularly useful in low light). If I didn't care about photography, I would be more tempted by Canon, particularly on their very high end bodies.

If, like most people, you care about auto-focus in video then you are going to have to mostly ignore the DSLRs. DSLRs focus very very fast in photography. And they are particularly good at tracking movement (which is why 99.99% of pro sports shooters use Canon or Nikon) However they mostly suck at video focus speed and this is especially obvious when shooting moving subjects.

The type of cameras which work well here are mirrorless cameras and cam-corders. The best of the bunch is probably the Sony a7s II, but it is over $2500 for just the body. Also exceptionally good would be the Panasonic GH series. Some of these would be in your budget if you buy used. The other option is the Sony SLT series, but they have a funny crop in video mode which makes your field of view very narrow (so you would have to invest in wider (costly) lenses).

Another consideration is recording length. In cinematic takes few recordings are longer than a few minutes. So if this is your planned shooting style, then ignore this. However for most hobbyists recording more then 30 minutes at a time is a big deal. The only way to get beyond 29 minutes is with either a cam-corder or a Panasonic GH series camera.

Lens costs. If you want to shoot quality videos, you will need quality lenses. Crappy lenses = crappy results. Everything hitting the sensor goes through the lens. Anything the lens does to the image will be very difficult to clean up in post processing. There are all-in-ones which look like a "mirrorless DSLR" but do not have a removable lens. Most of them are junk because the sensors are too small (meaning noisy, poor DOF control, etc). The two which are "ok" are the Panasonic FZ1000 and the Sony RX10 (and rx10 II). The FZ1000 has slightly faster focus speed. The rx10 has a better lens and has weather sealing. Both have "ok" 1" sensors. Meaning they are larger than other non-DSLRs but smaller sensor than the smallest one used in a DSLR. Neither of these can record longer than 29.5 minutes at a time. I use a Sony RX10 to record sporting events. There are enough pauses so I can stop and re-start the recording before ut times out.

For event videos I use a Panasonic GH3. You can get one used (from amazon and other places) for between 5 and 800 for the body only. Typical would be around 600. It varies depending on condition,shuttercount and bundled items. This has a m4/3 sensor and so you won't get the subject isolation or clean low-light images of a DSLR, but it is reasonably close. You will need to budget for a 14-42 lens with OIS. Used this is around 170, new 250. This lens will not get it done for you if you want quality, but you will need it for all the "fill in the blank" shots. This lens is not weather sealed. The sealed version is much more. You should also start saving for a stabilized "prime" lens (a lens with no zoom) such as the $600 42.5 f/1.7 from panasonic. Optionally you can spend 2-300 on a tripod and get Olympus 45mm f/1.8 for 260. This lens does not have stabilization and so a tripod is essential. You'll want a tripod regardless since the results from a tripod (or even a monopod) are simply much better than stabilization can provide.

For sports I opted for the Sony RX10 (first version). You can get them used from 4-800 depending on the same sorts of things as the gh3. Normal would be around 5-600. This has a 24-200mm f/2.8 lens. It won't get the subject isolation or low-light images of a m4/3 camera like the gh3, but for most uses it will be close enough. This i(and the newer versions) s the only all-in-in (Bridge camera) ones with "ok" image quality and weather sealing. The fz1000 is sin someways better and cheaper, but in low light it is let down by its lens in anything but bright light.