You are correct about the second scenario, for the reason you give, but not about the first scenario. The theory of the finite population correction (fpc) applies only to a random sample without replacement (Lohr (2009) Sec 2.8,pp 51-530. The key word is random. The hallmark of a random sample is that selection is determined by random numbers or the physical equivalent. In your first scenario the 45% of the population who responded were not selected by random numbers. The same would be true if the 45% were part of an even large random sample of the population: response is not governed by random numbers.
Even if you have a sample of a substantial part of the population with (near) 100% response, you should still omit the fpc if the purpose of your study is to develop predictions, estimate odds ratios, or to otherwise test hypotheses or quote p-values. The reasoning is interesting (Cochran, 1977, p.39): For a finite population it is seldom of scientific interest to ask if a null hypothesis (e.g. that two proportions are equal) is exactly true. Except by a very rare chance, it will not be, as one would discover this by enumerating the entire population. This leads to the adoption of a "superpopulation" viewpoint, which is taken by almost all statisticians these days. Your second scenario is a variant of this. See also Deming(1966) pp 247-261 "Distinction between enumerative and analystic studies"; Korn and Graubard (1999), p. 227.
ADDED NOV 26
I should have noted that the finite population correction is a minor concern here. The major problem is the 55% non-response and the subsequent non-response bias. Survey professionals universally agree that it is better to take a smaller manageable sample and to focus on reducing non-response by personalizing the initial contacts and by following-up with non-responders. Post-survey weighting fixes may also help, but will increase standard errors.
In summary, to answer your three questions:
- Your interpretation of the first scenario is incorrect.
- You really don't need to say anything. If your goal is to describe only the finite population from which you drew the sample, then you can mention that you omit the fpc because the effect is miniscule. Otherwise, when you do hypothesis testing or prediction, you could mention that omit the fpc, but I've never seen anyone do it.
- The decision of whether to use the fpc is the assessment you describe in the question.
So the answer is "Yes".
Additional discussion See a related CV discussion here.
References
Cochran, W. G. (1977). Sampling techniques (3rd Ed.). New York: Wiley.
Deming, W. E. (1966). Some theory of sampling. New York: Dover Publications.
Korn, E. L., & Graubard, B. I. (1999). Analysis of health surveys (Wiley series in probability and statistics). New York: Wiley.
Levy, Paul S, and Stanley Lemeshow. 2008. Sampling of populations : methods and applications. Wiley series in survey methodology. Hoboken, N.J: Wiley.
Lohr, Sharon L. 2009. Sampling: Design and Analysis. Boston, MA: Cengage Brooks/Cole.
Best Answer
The threshold is chosen such that it ensures convergence of the hypergeometric distribution ($\sqrt{\frac{N-n}{N-1}}$ is its SD), instead of a binomial distribution (for sampling with replacement), to a normal distribution (this is the Central Limit Theorem, see e.g., The Normal Curve, the Central Limit Theorem, and Markov's and Chebychev's Inequalities for Random Variables). In other words, when $n/N\leq 0.05$ (i.e., $n$ is not 'too large' compared to $N$), the FPC can safely be ignored; it is easy to see how the correction factor evolves with varying $n$ for a fixed $N$: with $N=10,000$, we have $\text{FPC}=.9995$ when $n=10$ while $\text{FPC}=.3162$ when $n=9,000$. When $N\to\infty$, the FPC approaches 1 and we are close to the situation of sampling with replacement (i.e., like with an infinite population).
To understand this results, a good starting point is to read some online tutorials on sampling theory where sampling is done without replacement (simple random sampling). This online tutorial on Nonparametric statistics has an illustration on computing the expectation and variance for a total.
You will notice that some authors use $N$ instead of $N-1$ in the denominator of the FPC; in fact, it depends on whether you work with the sample or population statistic: for the variance, it will be $N$ instead of $N-1$ if you are interested in $S^2$ rather than $\sigma^2$.
As for online references, I can suggest you