I was completely confused by that same explanation for a while. Don't even think of it like that. Think of it like this.
Definition: The effective rate of interest during the $n$th time period is
$$i_n = \frac{A(n) - A(n-1)}{A(n-1)}$$
Definition: The effective rate of discount during the $n$th time period is
$$d_n = \frac{A(n) - A(n-1)}{A(n)}$$
where $A(n)$ is the amount function (as defined in Kellison), the amount of money you have at time $n$. So, all you really need to understand here is that the rate of interest is a rate based on what you start with during the period. The rate of discount is a rate based on what you end up with. It's just two ways of looking at the same situation. There aren't a whole lot of real world situations where you borrow a bunch of money and immediately give some of it back. You would just borrow less.
By the way, that formula is all you need to calculate the effective rate of discount during period $n$ no matter what your $A(n)$ function is. So, in particular, it would work for your specific question of simple discount.
Question: Given a rate of 10% simple discount, calculate the effective rate of discount during period 5.
Answer: If we have 10% simple discount, then we know our accumulation function is $a(t) = \frac{1}{1 - 0.1t}$ for $0 \leq t < \frac{1}{d} = 10$. This is basically the definition of simple discount. If you have simple discount, this is your accumulation function. Memorize that. Then use it.
Therefore
$$d_5 = \frac{a(5) - a(4)}{a(5)} = \frac{2-10/6}{2} = \frac{1}{6} = 16.666666... \%$$
If you wanted to calculate the effective rate of interest when you are given the effective rate of simple discount, you can do that too. For example, in this same example,
$$i_5 = \frac{a(5) - a(4)}{a(4)} = \frac{2-10/6}{10/6} = \frac{1}{5} = 20 \%$$
Nothing changed. We're just looking at the same problem differently. In the discount case, how much money did we earn that period relative to how much we had at the end? In the interest case, how much money did we earn that period relative to how much we had at the beginning.
Your question doesn't really make sense. Since $t=0.5$, annual compounding is equivalent to simple interest since no compounding takes place. Also, the question is saying that $X$ is worth $4,992$ today, so you can answer (a) and (b) as
$$FV=X(1+0.5r)=4992(1+0.5*0.08)=5191.68.$$
(c) and (d) don't really make sense. When you're given a discount rate, you're usually computing present value, but this is already given to you.
It may help you to review the following formulas below the line.
Suppose we are investing an amount worth $X$ today (present value). Then we have with $t$ expressed in years and $r$ the interest rate quoted in the corresponding compounding frequency (otherwise known as the annual percentage rate or APR):
$$FV_{\text{simple}}=X(1+rt),$$
$$FV_{\text{m-compounded}}=X\left(1+\frac{r}{m}\right)^{mt},$$
$$FV_{\text{continuously-compounded}}=Xe^{rt}.$$
(Note for the $m$-compounding formula that $mt$ must be an integer, i.e. the time horizon $t$ must satisfy $mt\in\mathbb{N}$. In your question, you cannot substitute $t=\frac{1}{2}$ into this formula since $m=1$ and $1\cdot\frac{1}{2}=\frac{1}{2}\notin\mathbb{N}$. The simple interest formula must be used in this case.)
The effective yield is the APR adjusted for the effects of compounding. It is simply defined as the relative percentage gain (or loss) on your initial investment (i.e. the ratio of the future value to the present value minus $1$) over a period of $t=1$:
$$EY=\frac{FV(1)-X}{X}=\frac{FV(1)}{X}-1.$$
Depending on how the APR is quoted, this reduces to (respective to the formulas above):
$$EY_{\text{simple}}=r,$$
$$EY_{\text{m-compounded}}=\left(1+\frac{r}{m}\right)^{m}-1,$$
$$EY_{\text{continuously-compounded}}=e^{r}-1.$$
Best Answer
The annual effective rate of interest for year $t$, which we denote by $i(t)$, is the ratio of the amount of interest earned in a year, from time $t−1$ to time $t$, to the accumulated amount at the beginning of the year (i.e., at time $t−1$): $$ i(t)=\frac{a(t)-a(t-1)}{a(t-1)} $$ For the simple-discount method, we have $a(t)=\frac{1}{1-dt}$ where $d=2\%$ is the simple discount rate. Observing that $a(5)=\frac{1}{1-0.02\times 5}=\frac{1}{0.9}$ and $a(4)=\frac{1}{1-0.02\times 4}=\frac{1}{0.92}$ $$ i(5)=\frac{a(5)-a(4)}{a(4)}\approx 2.22222\% $$
NOTE: this exercise comes from the book Mathematical Interest Theory, 2nd ed. (Leslie Vaaler,James Daniel): ex. Sec 1.8 n. (2) at pag 65, sol. at pag. 441