Unfortunately mathematical terminology is not really standardized. To put it pointedly: To some extent it depends on the author's taste. Sometimes this has a geographic or cultural background. For example, there is a topological space known as the "Moore plane" in the United States and as the "Niemytzki plane" in Eastern Europe.
This especially concerns separation axioms in topology. It seems that there are no disagreements concerning the axioms $T_1$ (one-point subsets are closed) and $T_2$ (Hausdorff). However, the use of "regular" and "normal" is not consistent in the literature.
Let us focus on the concept of "normal". The undisputed core of this axiom is that disjoint closed subsets have disjoint open neighborhoods. One may definitely argue that this "closed subset separation property" deserves a name, and indeed some authors use the word "normal" for it (and do not require that normal spaces are $T_1$). However, a large and influential group of authors does not endorse this use of language and include $T_1$ (or Hausdorff, which amount to the same in this context) in the definition of "normal". Thus they do not have an own name for the closed subset separation property. Here are some examples besides Munkres:
Bourbaki, Nicolas. General Topology: Chapters 1-4. Vol. 18. Springer Science & Business Media, 1998.
Engelking, Ryszard. "General topology." Sigma series in pure mathematics 6 (1989).
Dugundji, James, "Topology." Allyn & Bacon Inc. (1966).
A nice compilation of separation axioms can be found in Wikipedia. This article was obviously written by a supporter of the "normal = closed subset separation property" fraction.
Also have a look into this Wikipedia article concerning the history of the separation axioms. Quotation from the section "Different definitions":
Every author agreed on $T_0$, $T_1$, and $T_2$. For the other axioms, however, different authors could use significantly different definitions, depending on what they were working on. These differences could develop because, if one assumes that a topological space satisfies the $T_1$ axiom, then the various definitions are (in most cases) equivalent. Thus, if one is going to make that assumption, then one would want to use the simplest definition. But if one did not make that assumption, then the simplest definition might not be the right one for the most useful concept; in any case, it would destroy the (transitive) entailment of $T_i$ by $T_j$, allowing (for example) non-Hausdorff regular spaces.
Topologists working on the metrisation problem generally did assume $T_1$; after all, all metric spaces are $T_1$. Thus, they used the simplest definitions for the $T_i$. Then, for those occasions when they did not assume $T_1$, they used words ("regular" and "normal") for the more complicated definitions, in order to contrast them with the simpler ones. This approach was used as late as 1970 with the publication of Counterexamples in Topology by Lynn A. Steen and J. Arthur Seebach, Jr.
In contrast, general topologists, led by John L. Kelley in 1955, usually did not assume $T_1$, so they studied the separation axioms in the greatest generality from the beginning. They used the more complicated definitions for $T_i$, so that they would always have a nice property relating $T_i$ to $T_j$. Then, for the simpler definitions, they used words (again, "regular" and "normal"). Both conventions could be said to follow the "original" meanings; the different meanings are the same for $T_1$ spaces, which was the original context. But the result was that different authors used the various terms in precisely opposite ways. Adding to the confusion, some literature will observe a nice distinction between an axiom and the space that satisfies the axiom, so that a $T_3$ space might need to satisfy the axioms $T_3$ and $T_0$ (e.g., in the Encyclopedic Dictionary of Mathematics, 2nd ed.).
Since 1970, the general topologists' terms have been growing in popularity, including in other branches of mathematics, such as analysis. But usage is still not consistent.
Munkres includes $T_1$ in the definition of "regular" and "normal", but you correctly observed that his use of "normal" in Exercise 6 of § 32 does not comply with his defnition. Nobody is perfect ...
Also have a look to Lemma 31.1 and Exercise 2, Section 31 of Munkres’ Topology and to Converse of Urysohn lemma. The latter suggests the suspicion that Munkres changed definitions from the first to the second edition. In § 31 he writes
It is clear that a regular space is Hausdorff, and that a normal space is regular
and adds (maybe in the second edition) in brackets
We need to include the condition that one-point sets be closed as part of the definition of regularity and normality in order for this to be the case. A two-point space in the indiscrete topology satisfies the other part of the definitions of regularity and normality, even though it is not Hausdorff.
Another well-known example for different interpretations is "compact". The core is that each open cover has a finite subcover, but many authors (e.g. Bourbaki, Engelking, Dugundji) include "Hausdorff" in the definition. Engelking uses the word "quasicompact" for spaces satisfying the "finite subcover" condition. Other authors (e.g. Munkres) do not do this and explicitly write "compact Hausdorff space" if necessary.
Conclusion.
Mathematical terminology is not as standardized as one might expect. Be flexible and attentive concerning definitions.
