Bingaloopa

Software Engineer and Software Developer are used interchangeably for the most part and I don't think you can really get any meaningful distinction just by the name (in most countries at least).

Some countries (Canada I think being one?) may require you be licensed/registered with an engineering body before you're allowed to call yourself an engineer, but thats not the case in most countries AFAIK

If the $2000 is to cover all equipment (and you have non already) then it's fairy low IMO. So you'll need to focus on the essentials first e.g a laptop/desktop

If you already have a decent computer though then think about the other things e.g a nice monitor, comfortable chair, good keyboard/mouse, headset etc

Personally I'd always say focus on comfort and making your work enciornemnt pleasant

I had the Sony WH1000XM3s and found similar - I still loved them but it was cutting out low droning noises etc that they shone at and voices definitely cut though quite a bit.

Recently though I got the Bose Quietcomfort Earbuds 2 (in ears), and the noise cancelling on them is next-level. So much better at cutting out higher pitches and I love them. I often use them in a room with a loud television and people speaking and they cut out a lot of speech and general noise

It still doesn't create silence in those conditions admittedly, but it's very impressive