The meaning of "spite" in "in spite of"

Hello. I would like to ask a question about the meaning of "spite" in "in spite of".

I have an english dictionary in which the meaing of "in spite of" is written as

in spite of: [If you say that sb did sth in spite of a fact you mean it is surprising that that fact did not prevent them form doing it.]

And I also found the meaning of "spite" in the same dictionary.

spite: [a desire to hurt, annoy, or offend someone.]

I can see that "spite" has bad and negative meanings. I think that following form of sentence is natural.

[In spite of (a bad thing, obstacle, difficulty, problem or disadvantage), I achieve (a good thing, goal or some desired result)]

For example, [In spite of his age, he still leads an active life.]

or [We are going to get married, in spite of the opposition of her family.]

But, I often see that the opposite form of sentence in various english texts.

[In spite of (a good thing), I achieve (a bad thing)]

For example, [In spite of his care, he slopped some water.]

or [In spite of many decades of investigations, some of the basic properties of ZnO still remain unclear.]

I do not think these sentences are natural because of the meaning of "spite".

Is there any wrong in my way of understanding? What is the proper usage of "in spite of"?