The numerator is the top part of a fraction, the denominator is the bottom part, and nominator is not an appropriate term for any part of a fraction.
I have seen nominator used to mean both "numerator" and "denominator". According to a question on this at English.stackexchange, this use of "nominator" is exceedingly rare.
Rather than people having been taught that "nominator" was appropriate, I think that it is far more often the case that the use of "nominator" is an eggcorn that has arisen due to its resemblance to the other two words.
I think using "nominator" should be discouraged because it already has a wholly different meaning, and has no etymological connection to fractions to speak of. It is also helps to confuse the meanings of the proper terms, if it is mixed with them.
I have never seen this claim in any textbook; in any case it's wrong. The claim seems to be that if you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ with $a < b$ and $c < d$, then $|a - b| < |c - d| \iff \frac{a}{b} < \frac{c}{d}$.
This is false. We can write $\frac{a}{b} = 1 - \frac{b-a}{b}$ and $\frac{c}{d} = 1 - \frac{d-c}{d}$, so you're comparing $\frac{b-a}{b}$ and $\frac{d-c}{d}$ (whichever is greater, the corresponding fraction is smaller). The first numerator may be smaller than the second, but the actual comparison of these fractions can of course go either way.
For example,
Here is one with $b - a < d - c$ but $\frac{a}{b} > \frac{c}{d}$: consider $\frac{2}{3} > \frac{3}{5}$.
Here is one with $b - a < d - c$ but $\frac{a}{b} = \frac{c}{d}$: consider $\frac{2}{3} = \frac{4}{6}$.
Here is one with $b - a < d - c$ but $\frac{a}{b} < \frac{c}{d}$: consider $\frac{2}{3} < \frac{5}{7}$.
So all results are possible; the test is nonsense.
Edit: Just for fun/completeness, here is a table showing pairs $(\frac{a}{b}, \frac{c}{d})$ with each possible combination of the two comparisions:
$$\begin{array}{c|c|c|c}
& \frac{a}{b}<\frac{c}{d} & \frac{a}{b}=\frac{c}{d} & \frac{a}{b}>\frac{c}{d}\\
\hline \\
b-a<d-c & \frac23,\frac57 & \frac23,\frac46 & \frac23,\frac35\\
\hline \\
b-a=d-c & \frac23,\frac34 & \frac23,\frac23 & \frac23,\frac12\\
\hline \\
b-a>d-c & \frac57,\frac23 & \frac46,\frac23 & \frac35,\frac23 \\
\end{array}$$
(If you want examples involving fractions greater than $1$, turn each of the fractions upside down. Each of the inequalities between the fractions will reverse direction, so you'll still have a complete set of examples.)
Edit: On looking at that segment of the video, it's possible (not very clear) that what he may have been saying is equivalent to the following claim, which is true: if you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ with $a > b$ and $c > d$, and if $a - b < c - d$ and $b > d$, then $\frac{a}{b} < \frac{c}{d}$. Proof:
$$\frac{a}{b} = 1 + \frac{a-b}{b} < 1 + \frac{c-d}{b} < 1 + \frac{c-d}{d} = \frac{c}{d}$$
With fractions less than $1$, the corresponding statement would be that if you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ with $a < b$ and $c < d$, and if $b - a < d - c$ and $b > d$, then $\frac{a}{b} > \frac{c}{d}$:
$$\frac{a}{b} = 1 - \frac{b-a}{b} > 1 - \frac{b-a}{d} > 1 - \frac{d-c}{d} = \frac{c}{d}$$
But these are so many conditions on the hypothesis that I wonder how often it will be useful.
Best Answer
The numerator and the denominator are the "parts" of a fraction.
It comes from the terminology of simple fractions. The fraction $\dfrac{2}{3}$ is two parts out of three parts.