I think hot weather alters firms' desire to sell at some given price because hot weather increases the demand for ice cream. (Mankiw's book)
Yes, you are wrong. As the quoted part explains,

In this case, economists say there has been an increase in “quantity supplied” but no change in “supply.” Supply refers to the position of the supply curve, whereas the quantity supplied refers to the amount producers wish to sell.

Are you willing to sell your current computer for say \$1 000? What is the lowest price that you are willing to sell for? (Just a ballpark estimate will do.)
If you came up with a number, you did that without knowing anything about the prospective buyers of your computer.
Of course it is very nice to be able to sell for more, if the customers are - for some reason - willing to paying more, but the ability to do this is determined by the equilibrium. The supply curve merely describes the vendors' preferences. What is the lowest price at which they are willing to sell / what quantity are they willing to sell if the price turns out to be X.