What is the difference between "Polynomial" and "Multinomial" in two or more variables? What is the difference between "Polynomial" and "Multinomial" in two or more variables?
Since, by definition:

Multinomial:
An algebraic expression having two or more (unlike) terms is called a Multinomial.

For example:
$5x^2 - 2x$ is a multinomial having $2$ terms,
$5x^3- 2xy + 7y^2$ is a multinomial having $3$ terms,
$7xy - 9yz + 6zx - 7$ is a multinomial having $4$ terms.

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.

Some books say "Multinomial" is one of the types of "Polynomial", and the other discuss it in particular.
Is the function $f$ cross "Polynomial" or "Multinomial"? Why?
$$f(x, y)=x y + y^2 + 2 x y^2 + y^3 - 3 x y^3 + x y^4,$$
 A: 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.
A: Under many definitions, polynomials and multinomials are synonyms (as commented above). Under any definition, your $f(x,y)$ is both a polynomial and a multinomial. A monomial is not a polynomial nor a multinomial. Poly $\to$ many, multi $\to$ multiple; neither multiple nor many can refer to one.
Polynomial is used more commonly though my experience is that multinomial is more commonly used when referring to plotted distributions. Whereas in describing a function polynomial is used almost exclusively.
A: The poly- of polynomial and the multi- of multinomial are, respectively, Greek and Latin prefixes meaning the same: many; and the -nomial is a Latin suffix meaning name. So both polynomial and multinomial just mean "many names". It is the interpretation of what "many names" is, precisely, and I don't think that there is a precise difference made that is agreed universally - you interpret by the context.
If you used the term binomial you'd expect a term such as $(x+y)$ in which there are two (bi-) names: $x$ and $y$; but you would not disagree that the term ${x^2+2x+1}$ was referred to as a polynomial but has only one name $x$, but this expression should be referred to as a monomial, but that term is rarely used.
My considered explanation for the inconsistency and confusion is in the interpretation of the suffix, -nomial, in which it should refer to name ($x$, $y$, ...) but some have interpreted it as referring to number (the indices).
A: Multinomials are generally distinct from polynomials.  You can't have a $1/x$ term in a polynomial, for instance. It may be that those terms have become synonyms through common misusage. 
I did see something interesting in a short little book POLYNOMIAL EQUATIONS: SYSTEMATIC THEORY SUMMARY, CHALLENGING EXAMPLES AND PROBLEMS.
A multinomial can often be converted to a polynomial by a simple variable substitution.
So if you have a multinomial of $1/x$, you substitute $z= 1/x$, then you have a polynomial in the variable $z$. Of course the new equation is then not valid at the singularity.
