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Are there differences in meanings implied by the following two formulations of interessieren and its noun das Interesse? Are there contexts where one would be used and not the other?
Ich interessiere mich an Geschichte.
Ich bin an Geschichte interessiert.
Ich habe Interesse an Geschichte.
Sentence 1 is wrong. 2. and 3. are correct.
The preposition für must be used with sich interessieren: Ich interessiere mich für Geschichte. Alternatively one can say Geschichte interessiert mich.
With interessiert sein and Interesse haben, an (with dative) must be used. Here, für would be incorrect.
The meaning of all three is practically identical.
But in other contexts, there can be a difference. [Ein] Interesse haben an and interessiert sein an can be used to express objective interests, as in "Das Land hat ein Interesse an einem freien Welthandel" ("The country has an interest in free world trade"). Sich interessieren für doesn't convey this meaning.
"Sich für etwas interessieren" has also a more personal meaning. If you say: "Ich interessiere mich für Deutsche Sprache" (or "Deutsche Sprache interessiert mich"), that means you have some sort of love for it. Like for a Hobby. Because "ich interessiere mich" basically means "I make myself interested in" (or in the second case "German makes me interested"). Whereas "Ich habe Interesse" is more of a momentary need for it or comes from reasoning. "Ich habe Interesse an Deutscher Sprache" sounds more like someone wants to learn German for business reasons. So it is more a thoughtful interest, a result of a process. "Ich habe Interesse" just means, that you have interest. "Ich interessiere mich" tells that the interest is intrinsic. This gets more obvious, as soon as you take the sentence "Ich interessiere mich für dich", which is a nice thing to say. I can't imagine someone saying: "Ich habe Interesse an dir". That sounds a bit like the other person is Amazon stock. However, you can often read "Er zeigte Interesse an ihr" or "Er hatte Interesse an ihr". In this case, you are talking about someone else's interest. So it is not yours and then "Interesse haben an" makes perfect sense, because you cannot judge whether it is intrinsic or from reasoning or even with bad intentions.
One more example. Now we negate the sentence: "Ich habe kein Interesse an Deutscher Sprache" is very neutral. "Ich interessiere mich nicht für Deutsche Sprache" can easily be interpreted as "I don't give a damn about German language". Once again "sich (nicht) interessieren für" is more emotional. On the other hand, if you have to reject someone, you better say "Ich habe kein Interesse an dir", which is rather neutral. If you say: "Ich interessiere mich nicht für dich", that's rude. Same for "Du interessierst mich nicht". Very rude, too.
If you use the verb (sich) interessieren, directly, the correct preposition is für. Another such verb is schwarmen für (to be enthusiastic about).
The correct preposition is an, if you use the idea of "interessieren" indirectly with a helping verb and a noun, "Ich habe Interesse an Geschichte," or a helping verb and an adjective, "Ich bin an Geschichte interessiert."
