That doesn’t mean energy has a weight.

No, it literally does mean that. If you put light in a box of mirrors the total weight of the box will literally increase by an amount equal to the energy of the photons. If you put some radioactive material in a theoretically perfectly sealed box from which no heat or light could escape, and weigh it while it decays into radiation, the weight will not change.

This applies to all forms of energy. A spring is heavier when compressed. An object gets heavier when you spin it, or heat it up. Sunlight hitting the earth most definitely makes it heavier. In fact, the sun hits the earth with about 4.4*10^16 watts of power, corresponding to about 0.5 kilogram per second.