It's not opposite but it is alittle off. I don't put any greens in the warm color category. And I usually don't put any purples in the warm color category.
To make it really simple, yellows, oranges, and reds are on the warm side.
Greens, Blues, and Purples are on the cool side.
Now is there a grey area? Say like Chartruse, I would put it on the cool side but there is definitely room to put it on the warm side. Especially if the color is very "intense." (that's the saturation of color). On the other hand, if the chartruse is either tinted (white is added to it) or it is dulled (black or grey is added to it) it will be much cooler.
This is an area that I don't believe I have discussed yet. It's where color can get really confusing. As if that is possible.
Intensity is just how bright the color is. An example of the brightest colors are those that are a pure color. Nothing is added. Some examples of these are the Primary Colors as they are often used in childrens quilts, "Gem Tones" such as: Emerald Green, Saphire Blue, Ruby Red etc is another example of intense colors. These pop as I'm sure I don't have to point out. Tints and Dull colors on the other hand receded into the background. A tint is any color that white has been added to, to decrease the intensity of the color. Pastels fall into this category. Dull colors have had black, or grey added to them for the same purpose. It decreases the intensity of the color. Examples of this would be most of the olive greens, the navy blues, and the almost black purples.
Art, as in everything else, you are going to find differing opinions to some extent. When that happens to me I evaluate the information and see what seems most correct to me and go with that. You will also find mistakes on the internet from time to time. I'm glad you asked me about the differences you found. Here is another difference you will find as you explore color wheels. The ones that are for Light rather than Paint have different primary and secondary colors. When my second husband took photography we had a major row about that until I researched it and found out that they are different. That really suprised me. For fabric use the color wheel used for paint.