Why do people insist to use the term "multilayer perceptron" instead of "multilayer perceptron network"?
Personally, I've seen "multilayer neural network" used more commonly than "multilayer perceptron".  If you like the former term better, use it.

That said: multilayer perceptron is an accepted term of art.  Your criticism is not unreasonable, but it is what it is.  Language exists to help us communicate... so pick terms that will help you communicate your intended meaning to your intended audience as effectively as possible.

If it helps you come to some kind of acceptance for this usage, you can think of "multilayer perceptron" as a single term that has a particular definition: don't think of "multilayer" as an adjective modifying "perceptron", but thik of "multilayer perceptron" is a single atomic phrase that has a particular definition.  For example, some mathematics textbooks define the term "ring" in a way such that a "commutative ring" is not actually a "ring"; that's just how it is, so just go with it and don't let it throw you.  As another example, an "algebraic integer" is not an "integer"; it's a complex number with some special properties.  This kind of situation appears all over the place in mathematics.

Ultimately, it's just a convention.  And that's ultimately what language boils down to: a convention.  These conventions sometimes can look a bit inconsistent or "improper", but if they are widely understood, then they still serve the ultimate goal, of clear and concise communication among folks who are knowledgeable in the field.

Bottom line: don't let a sense of what is "proper" overcome the real goal.  Keep in mind the real purpose of these kinds of terms: namely, effective and efficient communication.