I don't see anything "unnatural in meaning" in any of your sentences. But let me point out one common mistake here. The word accidently does not exist in English. The adverb is never formed by adding -ly to the noun (= accident); it is constructed by adding the adverbial suffix to the adjective (= accidental). Thus the correct spelling should be accidentally .
I accidentally broke the vase.
I broke the vase accidentally.