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Sometimes, the wording of a law or contract is vague and imprecise, and it allows people to abuse it, but doing so goes against the spirit of the law or the contract when it was written, or in other words the intention the people who wrote the law or contract. Is there a word for this?

This is known as a loophole.

There is a principle in American jurisprudence that if a law is too vague, it is not valid. Similarly, if a contract allows more than one reasonable interpretation, the party who wrote the contract does not get to decide which interpretation(s) will be used. Instead, the other party gets to choose.

Jasper's suggestion of "loophole" is excellent, but you may also hear this situation arising from unintended wording referred to as "a technicality".

Consider the etymology of loophole:

also loop-hole, mid-15c., from hole (n.). + Middle English loupe "narrow window, slit-opening in a wall" for protection of archers while shooting, or for light and ventilation (c. 1300), which, along with Medieval Latin loupa, lobia probably is a specialized word from a continental Germanic source, such as Middle Dutch lupen "to watch, peer." Figurative sense of "outlet, means of escape" is from 1660s.

Loopholes are placed intentionally for avoiding certain provisions, usually for benefit of special interest groups or other political donors. Those who take advantage of loopholes are using the law as it was intended, at least on some level. The term can also be problematic because people sometimes (especially in political contexts) disparage aspects of even precise laws that they simply dislike as loopholes, e.g., the “gun show loophole” or the “mortgage interest loophole,” and the discussion degrades into an argument over whether it is truly a loophole or part of the intent.

The scenario from your question involves twisting or perverting the law into saying something that it does not say. Read on for more more accurate descriptions.

Applied to rhetoric in general and not just legal argument is sophistry:

The use of clever but false arguments, especially with the intention of deceiving.
‘trying to argue that I had benefited in any way from the disaster was pure sophistry’

Another possibility is legalism:

Excessive adherence to law or formula.

Realize that the term can have religious connotations, as described in the second sense of the linked definition.

For a more cynical and even inflammatory characterization, Lavrentiy Beria was head of Stalin’s secret police and is said to have boasted, “Show me the man, and I’ll show you the crime.” In other words, he believed he could manipulate the law to find any person guilty. Crisis Magazine ran a piece called Beware the Beria Method (emphasis added). Note that this is not in common usage.

In Federalist No. 41, James Madison rejected the claim that the then-proposed U.S. constitution conferred to the new federal government any power that might be alleged to benefit the general welfare with (emphasis added)

No stronger proof could be given of the distress under which these writers labor for objections, than their stooping to such a misconstruction.

Finally, someone seeking to abuse law or contract terms is said to use a strained interpretation, an overly literal interpretation, or a tortured interpretation. A less formal and broader way to state it is the person is out in left field or, to avoid the personal attack, that the interpretation came out of left field.