Translation:It is not cool today.
cold is more... near the freeze. cool is only a "fresh" temperature
I am not an english native speaker, I may be wrong.
You're correct. You could say that "cool" is tolerably low tempurature, something that you can wear only a sweater for. "cold" is much more biting, such that you need a coat and thick pants.
You generally enjoy cool weather.
If it's cold you'll usually want to wear warm clothes.
I think the meaning of すずしい is closer to something like pleasant chill, like it's nice weather for taking a stroll.
...cool as in English colloquial? i.e. "nice" or "fashionable"? (Duolingo messed up my original post!)
I'm not sure but I don't think so. I think the Japanese word for "cool" is 格好いい(かっこいい).
Would it be correct to say "on this day" instead of "today" in this sentence? (It is not cool on this day / On this day, it is not cool)