What is the difference between "Polynomial" and "Multinomial" in two or more variables?

As you mentioned,

Polynomials in two or more variables:

An algebraic expression in two or more variables is called a Polynomial if the Power of every variable in each term is a whole number.

Now the crucial point is that polynomials can be classified as monomial ( 1 term ) , binomial ( 2 terms ), trinomial (3 terms) , quadrinomial (4 terms), quintinomial (5 terms), multinomial ( polynomial having more than one terms ) etc depending on the number of terms present in their expressions.

So multinomial is a type of polynomial having more than one terms in it.

Therefore, $$f(x, y)=x y + y^2 + 2 x y^2 + y^3 - 3 x y^3 + x y^4,$$

is a multivariable multinomial polynomial i.e. it is a polynomial having more than one variable and more than one terms.