"More likely than not" would be the 'balance of probabilities' or 'preponderance of the evidence' standard, as used in civil cases, correct.

However, "Beyond a reasonable doubt" includes a probability too, within the word 'reasonable'. That is, a probability or confidence with no room for reasonable doubt. That very much does not mean a certainty, or that there's zero room for any doubt whatsoever.

the only way one could convict is if that story is so preposterous that it doesn't create even a possibility of doubt in your head.

This is a common misconception, but it's simply untrue. Look at pretty much any criminal case where the defendant actually goes to trial--you can claim many things which are less than preposterous which may not be believed by the judge or jury. Look at any number of he-said-she-said decisions on crimes against the person, any number of potential reasons someone might claim they were found with contraband which totally isn't theirs, any number of cases where someone attempted to plead insanity or sleepwalking or some other defense. These may not be facially preposterous, but don't automatically outweigh the reasonable doubt standard.