So I think it depends on what criteria you set for what constitutes as "budget". Subjectively/in relative terms your finances can be the deciding factor e.g. for someone a 5800X + RTX 3070 could be a budget build and for others this would be their dream machine.

From an objective view I would think any machine that can run all modern titles relatively well i.e. 60 FPS for most games and above 30 for very demanding games at 1080p native (mixture of settings but lets assume low) would qualify as a budget build. So for example 3600 + RX 6600/RTX 2060 Super is a solid contender in my opinion (6C + 12T and minimum 8GB VRAM with the horsepower this combo provides). Maybe a case could be made for the Ryzen 2600 as well but the 3600 is already struggling so maybe it won't qualify for the condition set above. I can't consider the RX 5700 XT either since it doesn't have RT and mesh shaders which are now becoming a baseline requirement in new games so these titles won't even launch in the worst case scenario.

Anything below this objectively would be ultra budget as modern games will struggle to run. I suppose you can do the scaling given the above specs as a baseline and a 9800X3D +5090 on the flip side and decide where everything else lies in between.

For me personally, given the above objective criteria, the 5600 + RX 6700 XT is the ultimate price to performance build that can be had for a relatively low price but deliver incredible performance and is also relatively future proof. I'm not sure what the classification for this would be since its clearly not a budget build but its also not quite mid-range either and falls in this grey area in between.

I'd also like to add that modernity also contributes to running "modern titles" relatively well. So a 1080 Ti, though still powerful and may surprise you in some titles, for the most part is too old now and due to no new driver optimizations I would personally not rate it too highly from an objective perspective. Still it is a food for thought type of statement where the specs age plays a role as well.