Question on significance of different ways of measuring Price Elasticity of Demand
Ideally the points $(x_1,y_1)$ and $(x_2,y_2)$ are very close. The assumption is that that they are so close that the elasticity of the demand function on the curve between them is nearly constant. This is what elasticity is trying to measure, local changes, because on the far side of the range of the demand function consumer behavior may be unobserved and totally different. E.g. to my knowledge no one has tried to sell a bottle of Coke for \$40,000 so we don't know what value the demand function would take. If you would choose a very distant point $(x_3,y_3)$ instead of the local $(x_2,y_2)$ you would get a very different value of elasticity.

So what does the book mean by initial value? Ideally the elasticity measured in points $(x_1,y_1)$ and $(x_2,y_2)$ would be constant because they are very close. Of course in practice this will not always be the case, meaning
$$
\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1} \neq \frac{(x_1-x_2)/x_2}{(y_1-y_2)/y_2}.
$$
Instead of taking one of these values as the elasticy of the curve we can improve our estamite by taking the arc elasticity which usually yields a value between these two.