Belief is not equivalent with intent in law.
Where the required mens rea is intention, it is intention that must be proved, not mere belief in a state of the world.
However, certain beliefs can themselves be relevant to the issue of intent.
For example, evidence that X held the belief that a particular powder could kill a person, when combined with evidence that X put that powder in a person's drink, could be significant evidence of intention to kill. This is the case even if it were to turn out that the powder is innocuous.
And to answer a question in the comments about how motive and intent are different in criminal law, the Supreme Court of Canada has explained:
In ordinary parlance, the words "intent" and "motive" are frequently used interchangeably, but in the criminal law they are distinct. In most criminal trials, the mental element, the mens rea with which the court is concerned, relates to "intent", i.e. the exercise of a free will to use particular means to produce a particular result, rather than with "motive", i.e. that which precedes and induces the exercise of the will. The mental element of a crime ordinarily involves no reference to motive. ... For example, D intends to (a) put poison in his uncle's tea, (b) to cause his uncle's death and (c) to inherit his money. We would normally say that (c) is his motive. Applying our test of "desired consequence" (c) is cerÂtainly also intended. The reason why it is considered merely a motive is that it is a consequence ulterior to the mens rea and the actus reus; it is no part of the crime. If this criterion as to the nature of motive be adopted then it follows that motive, by definition, is irrelevant to criminal responsibility—that is, a man may be Iawfully convicted of a crime whatever his motive may be, or even if he had no motive.