No, it doesn't necessarily have a negative connotation, though it can have.
Meaning 1 is usually shown by words such as against or towards. Meaning 2 (which I think is rare) takes no object, or maybe an infinitive
Some of the examples from the OED (which mentions both meanings, but does not treat them as different senses):
I hope you will now, despite your rather evident animus against her, set this to her credit. (1911, meaning 1)
They have an instinctive appreciation of the animus that actuates the policy of a foreign country. (1867, meaning 2. All the examples newer than this have against.)
Having said that, the example from Thackeray (1840):
From the animus with which the case has been conducted,..it was easy to see the result.
does seem to have the first meaning, even though there is no explicit object.