During these 'in between' times the Earth's axis is neither tilted towards nor away from the Sun. This means that all places on the Earth see the Sun at an 'average' sort of height in the sky. So why not use the same name for both of these times of year? We could have the cycle of seasons as Winter-Average-Summer-Average-Winter.
The problem is that the average season that comes after winter is very different from the average that comes after summer. In the places that have just had summer the Earth is just beginning to cool down as they head for winter. In the same places after winter, the Earth is just beginning to warm up as they head for summer.
So what we really have is Winter-Average warming-Summer-Average cooling-Winter.
As you realise, since they are really very different from each other, we have given nicer names to the two average seasons. We call them Spring and Autumn.