I think 1) and 2) are both right, although 1) would be used more. I think 1) is not grammatically right because think is a verb and a verb cannot be modified by adjectives like positive. However, it is widely used and that’s why we cannot say it’s wrong. What do you think?
I think you’re right. “Positively” would be more logical because it must be an adverb. However, it is a “fashionable” expression, so that those who use it probably use it colloquially, where grammar is not very strict. To “think positive” is said often indeed.