It’s exactly as you said—different clouds have different effects on the climate. Some warm things, some cool things.

It’s more complicated than this, but the way I learned in school was that all clouds reflect sunlight, but they can also block some IR from leaving the earth. Low clouds absorb IR from the surface but re-emit it at a similar level to the ground because they are a similar temperature. In this case the reflective effect dominates, so most low clouds have a cooling effect. But high clouds are very cold—this means they absorb the same IR from the surface but emit far less into space, keeping the planet warmer. This dominates the reflective influence, so high clouds can warm the climate.

Clouds at night are particularly warming, since they don’t reflect sunlight. For this reason, redeye flights have a greater warming effect than other flights. Supersonic jets and other very high flying aircraft can also be extra harmful.

My uneducated guess - That article mentions the high sulphur content causing them to be reflective. If those weren't there, the sun would be hitting the dark water and being absorbed instead of reflected. Contrails are just water and so probably absorb more energy instead of reflecting.