I would first and foremost consider a few generations old flagship or prosumer model from a reputable used seller like usedphotopro or mpb. They'll check over the units, give it a grade on quality (sometimes cosmetic blemishes will get you a great deal) and they will give a warranty for everything for a few months as part of the deal.

With interchangeable lens cameras you're mostly buying into a lens system so think along those terms. ie... if you buy a canon you have to buy lenses that fit on canon's mount and sometimes there are limitations for 3rd party glass that canon, for example, lock out if you don't buy their first party stuff.

Canon's prosumer DLSRs can be gotten for a good price and have a lot of options. Things like the 7Dmk2 or the 5Dm4. IDK about Nikon. Personally I would also say to take a look at micro four thirds cameras. OM systems (Olympus) you can get a great deal on some of their stuff that works really well and will be feature rich (some really great optical and in body image stabilization), and the m43 system being physically smaller you will have a much lighter kit to carry around. I picked up an Olympus OM-D E-M5 Mark II and I use it for bird photography regularly. I'm probably in maybe $700 with a camera, 3-4 lenses, new batteries and a bag for that whole setup. Olympus and Panisonic (Lumix) cameras both use the m43 mount (since it's basically open source) so lenses are mostly interchangeable and there are lots of third party lenses... also they're usually cheaper than larger system lenses like canon and nikon. The RAW files I get from my OM camera are much smaller than the ones I get from my canon which is also nice.

Since you're talking about birds, I would suggest considering m43 and aps-c because of the "reach" you'll get from telephoto lenses due to the extended crop factor. I do a lot of my bird photography with the OM camera and an Olympus m.zuiko lens that's 40-150mm... but because of the 2x crop factor on m43 it makes it basically an 80-300 and because it has IBIS and it weighs nothing I makes for good stable not blurry hand held shooting.

Plan a bit but don't spend a bunch until you know what you are actually finding a use for. I have a bunch of lenses but I mostly find myself using that 40-150mm and my fast 45mm portrait lens on that camera. Since you're talking about bird photography then a zoom that gets you into a 300-400mm telephoto equivalent is a really good starting point. That's a good amount of reach while still being able to hand hold without a lot of blur and still able to focus on things closer. If you find scenarios where things are too far or too close, that is when you can start considering a new lens.