There are many things to consider and mixed up units and babble words for easy preys...
Pick some cameras, rent them, try them, decide.
Reading further may help deciding where to start ruling some choices straight out and shortening they tryout period.
Look up this page on digitalcameraworld. It shows a relation between focal length (in mm) and sensor size (in mm), and view angle. The view angle is critical for you because it defines how far from the object you have to be so it covers the portion of scene you want. If you want to crunch numbers the tangent of half the angle equals to the width divided by the distance.
The camera sensor size and resolution seems to be the defining parameters but they are, actually the last ones to be taken in consideration.
The critical parameters are sensor noise and lens so-called speed, more to it later.
We shall start fom answering most crucial questions:
- How fast is your target about to move?
- How dark the scene is to be expected?
Answer to those questions define the lens you are looking for in term of its "speed". This property is defined in F-stop units like F3.5, F1.8, F8 and it describes how wide the lens' aperture can open. In other words it describes how much light can pass through the lens when aperture is wide open. The lower value the better. How is the speed value derived and how to calculate with it is way beuond the scope of this answer.
The faster your target moves and the darker the scene you expect, the lower F-values (faster lens) you are looking for.
The answers also define what sensor you are looking for in terms of sensitivity and noise. To get the picture bright enough you need to get enough light to the sensor or brighten the image artificially - multiply all the values by some ammount. You can clealy see that artificial brightening increase the noise. If you want to bring more light to the sensor you either need to open the apreture or expose the sensor for longer time. And here we are directly in the core of the questions.
Now we know what angles of view we are aiming for and we broadly have idea how fast the lens should be (as fast as buget allows, of course! no joking).
We can now move back to the diagrams on digitalcameraworld and check the lens prices for the common sensor sizes. Again, if you want to crunch number you can look for exposure equation - it's a function of shutter speed, aperture, sensitivity and scene brightness.
While you are checking the lenses you can also check the camera bodies for their ISO sensitivity. The higher sensitivity it allows (at noise acceptable for you) the easier you can go on the lens speed.
This is all but theory so far and we all know, theory is grey, green is the tree of life.
Pick couple of builds from different camera manufacturers (Canon, Nikon, Sony, Panasonic, Konica-Minolta, Pentax,...) lens manufacturers ([camera brand], Sigma, Zeiss, Tamron,...) and body types (DSLR, mirrorless,...) and sensor sizes (3/4, APS-C, fullframe, 6cm, ...) and keep the build within your budget limit times 1.2 - 1.4. Trust me, you will end up over-spending sooner or later, better to accept it sooner than later. Do not cross out second-hand gear! You can save quite a lot by sacrificing the feel of opening the not-yet-opened seals.
Here you can start thinking about sensor resolution, connectivity and other factors, but they doesn't matter that much.
You have your picks now and you narrowed the pool in, say, 6-10 options.
Rent one or two picks for a weekend and go out shooting (with your current gear too). If you are to shooting birds you ca pretend they are there and shoot the backgrounds instead. The aim is to get your hand on the cameras, try them out, push your limits with them. And write notes about it. How do you feel about them, how easy it is to operate them... Then rent another pick...
You may realize you want something slightly different, so change it.
Then you will end up with a table of YOUR assessments and prices. And this table is the one to decide on, because it is YOU taking the pictures, not anyone of US and I'm sure OUR table will be different from YOURS.