The length of a year is, in our world, roughly defined by the passing of four seasons.

That's so wrong! Only mild Western Europe has nice four seasons, most of the world has two seasons and the tropics stay the same all year.

How do they define a year

12 full moons = one year.

That's a lunar year, a solar year is one revolution around the sun. The Christian year is solar, the Islamic year is fully lunar, the Jewish year has lunar months, but every three years they add an extra month to stay inline with the solar year, that's why Passover & Easter move while Christmas (basically a Roman invention) is fixed.

There is a verse in the Quran (9:36): "The number of months are twelve in the sight of God" and the word month in Arabic is actually connected to the full moon (literally: to appear high & bright). That's the most simple way to calculate a year because you don't need any Astronomy or correction, but it is quite useless for use in Agriculture that depend on Solar calendar to know the planting season, etc., but you need Astronomy for that, which is why the seasons are fixed by the Maesters of the Citadel, but not the year.

Nowhere in the text is it explicitly mentioned that the years and the months are lunar, but all time measurement beyond a week is done with moons, so we can deduce that the year is lunar.

Instead of seasons lasting several years?

I have already elsewhere wrote about this: basically we can define a Seasonal year as a year from the white raven of winter to the next one. From 274 AC to 300 AC there were 8 to 9 such years, giving us an average of 3 Lunar years per Seconal year, that's just an average of 9 months per winter.