There are (at least) four different definitions of winter:

• Astronomical seasons are what you describe, and these are obviously based on astronomical events, for which the equinoxes and solstices are the perfect fix points. That the seasons start at these days is purely conventional, and in some times (roman empire) and places (Celtic calendar) people used these days as the center point of the season instead (also known as “Solar seasons”). Why those don’t match up with the actual temperature has already been explained in countless other replies.
• Meteorological seasons use a simple, month based approach, where winter is just December, January and February. That makes it easier for statistical usage but obviously is also just a man-made convention.
• The energy sector defines winter as the time of extended energy needs due to heating. Where I live, that’s defined as 1. of October - 31. of March, but for obvious reasons that’s highly dependent on where you are.
• Phenological seasons are a bit more what you’re looking for: They are based on biological events in indicator plants and a lot more complex than just being four fixed periods and only ever the same for small regions. Where I live, there are ten phenological seasons, winter starts when the English Oaks drop their leafs and ends with the blooming of the Common Hazle.

In other cultures there might be vastly different seasons. In many tropical countries you’ll only have the dry season and the monsun season.