First, with the reality-check tag, I'm not sure you could have a 4-year orbit around a pair of stars and still be within the "Goldilocks Zone" of the system.
Here is a rough set-up that could allow multiple groups of seasons within the same solar year. This does require the suns be far enough apart for their mutual orbits to have a period a significant fraction of the orbital period of the planet. The set-up is much easier to visualize if the periods are in resonance so that each planetary year matches up with a set configuration of the suns during its orbit. Distance from the suns may have a slight effect on seasons, but the primary driver will be the regular eclipsing of each sun by the other. I believe this would be most visually distinctive with two suns of about the same size, but of slightly different colors.
Your planet is in a significant elliptical orbit around the stars. Start its orbit at the apogee and sync up the suns so that one is eclipsing the other (only one sun in the sky). Also have the tilt of the planet align with the suns. For the hemisphere tilted away from the suns (call it north), this will be your coldest time of the year. The hemisphere tilted towards the suns (south), though technically in "summer", with only a single sun in the sky it will be relatively cold.
Move the planet 1/8 of its orbit. There are now two suns in the sky at their farthest visual distance from each other. This is summer. Two suns in the sky warm the planet more than a single sun, regardless of axial tilt.
Move the planet 1/8 of its orbit. The suns are in eclipse again and winter is back.
Another 1/8. This will be the "hot" summer for the north with 2 suns, closer orbit, and tilting towards the suns.
Another 1/8. The perigee of the orbit, with the north tilted directly at the suns. Luckily the suns are again in eclipse, so it is winter, but a "warm" winter.
The next 4/8 will be a mirror of the first, moving back to the apogee.
You could introduce additional differences by having one star emit more infrared or ultraviolet and note the differences that would make when one would eclipse the other.