The money question first, because it explains your cheap one: most of a budget tripod's failure is the head, not the legs.
A head that creeps under load, or that shifts when you tighten it, means your composition moves after you set it and the camera is never quite still. People blame the legs, buy heavier legs, and have the same problem.
So the sensible split is to buy the head deliberately, separately if you can. A decent ball head with a proper clamp is transformative and is often the cheaper half of the purchase.
On the legs, what actually produces stability:
Fewer leg sections. Each twist or flip lock is a joint that flexes. Four sections pack smaller and are less rigid than three.
Thicker tubes, particularly the bottom section, which is the thinnest and does most of the wobbling.
Not extending the centre column. The centre column is a single unbraced tube and raising it undoes much of what the legs provide. Treat a tripod's real height as its height with the column down.
Carbon over aluminium if you are carrying it, same rigidity for less weight, and it damps vibration better. This is where the price is, and for a tripod you will actually carry it is the money worth spending.