Well, no - taxonomically speaking, apes, humans and lemurs etc are not monkeys. They are all primates. Culturally speaking, you're right, people get them confused all the time, to the point that it may become the accepted definition in some places. But academically speaking, globally, monkeys are a distinct group of primates. Defined by having a tail, or the stub of a tail.
Lemurs are not monkeys, yes, as monkeys refers to simians. Again though, in the taxonomic world, animals are not defined by traits anymore, but by their evolutionary history and shared ancestry. Old World monkeys share a more recent common ancestor with apes than with New World monkeys, which means excluding apes from being monkeys creates an unnatural grouping. Excluding apes from a group within which they originated just because they lack tails is like excluding leatherback sea turtles from being reptiles because they aren't cold blooded.
And it's definitely not a global thing, many languages make no distinction between monkeys and apes. English just decided to separate them based on an arbitrary characteristic, and probably because it considers humans and other apes to be "more advanced" than mere monkeys.
In fairness, "monkey" was never meant to be a scientific term - the equivalent name for the group in taxonomy is simian. It's a common name, and common names don't have to adhere to strict phylogenetic definitions. However, it does mean that calling apes monkeys is technically correct, it's just a way of saying "simian" in laymans terms.
Wikipedia is not a reliable source.
While yeah, anyone can edit it, Wikipedia is a lot more reliable than secondary school teachers would have you believe. Popular pages (eg monkeys) are reviewed thousands of times, to the point when they're arguably even more trustworthy than peer-reviewed scientific papers:
At a time when it’s increasingly difficult to separate truth from falsehood, Wikipedia is an accessible tool for fact-checking and fighting misinformation.
theconversation.com
For me, the way it's based on citations just makes it an easy resource to show the current thinking on a particular topic.