But the word מָלַךְ is the same for all of these kings - wouldn't all of the מָלַךְs either refer to the beginning of a co-regency, or the beginning of a reign alone? And if not: how do we differentiate the beginning of a co-regency from the beginning of a reign alone, based on the words used or the context?
I am not sure it is really possible to understand the nature of the reign of each king merely from the context in close vicinity to the verse(s) referring to the start and end of the reign. One of the major issues in studying Tanach is trying to understand things from the POV of the writers and their original audience (we, later readers, are a secondary audience). In other words, since the writers (typically prophets) lived in a certain time period and were directing their words at certain people, there were naturally things left unsaid, or at least, not clearly emphasized, because all of these people already knew all of these things. There are things that were accepted as social conventions, and therefore there was no need to state them.
For example, there is a thesis in Hebrew that I have read part of, called "Reconstruction of the Chronicle Used by the Author of the Book of Kings in Light of the Mesopotamian Chronicles", by Dr. Yosi Baruchi. The thesis attempts to reconstruct some of the bibliography that the author of Sefer Melachim may have based himself on. Post-reconstruction of the sources, Dr. Baruchi attempted to reconstruct the chronology of the list of kings. As he himself stated, the differences between the MT (Masoretic Text) and the LXX point to (somewhat) bad attempts by the LXX translator(s) to correct the perceived chronological issues in the book (p. 151 and n. 1). However, in order to properly understand the MT's chronology, there are a few assumptions that must be accepted:
- The years of the kings, both of Yisrael and of Yehudah were not counted from the start of the reign but from Rosh Hashanah [La'melachim] (the New Year [of the Kings])1.
- Rosh Hashanah in the two kingdoms was not at the same time: For the kings of Yehudah it was in the spring while for the kings of Yisrael it was in the fall.
- Another key difference between the two kingdoms: The time that passed between the crowning of a king of Yehudah until the first Rosh Hashanah was counted as part of the reign of the previous king, while in the kingdom of Yisrael it was counted as part of the reign of the new king. (pp. 151-152 and related notes)
These are rules that have been concluded from analyzing the lists in their entirety, as well as perhaps other portions of this book (consider, for example, Melachim 1:12:32-33 which shows that Yerov'am changed the calendar of the kingdom of Yisrael).
Even if these rules are incorrect or not entirely correct, I believe they give us the understanding that there were rules, and very likely there were different rules in each kingdom, and whoever lived at the time likely knew these rules and did not need to have them included in the book. We find, therefore, that behind the term "מָלַךְ" were all of these different factors that defined the length of the reign, but since it was obvious that were differences between reigns depending on various factors, there was no need to write anything other than "מָלַךְ".
I hope that clarifies things.
As a side-note, I found today in the library an edition of פירוש רבנו מיוחס ב"ר אליהו on Chronicles and there he states (at least once; I haven't gone through the entire commentary yet, short as it may be) on Chronicles 2:16:1 (emphasis mine):
"שלשים ושש למלכות אסא. אי אפשר לומר כן שהרי בעשא מת בשנת עשרים ושש לאסא כמו שמפורש במלכים ולא נכתב אלא לדורשה. וכן דרשו רבותינו כנגד שלושים ושש שנה שנתחתן שלמה לבת פרעה כנגדן נגזרה גזירה על מלכות בית דוד לבטל ועוד תחזור להם וכנגדן שלשים ושש שנה שנגזרה על מלכות ארם שתמשול בישראל ועוד תפול ביד בית דוד."
In other words, in his view, whenever there is a chronological discrepancy, that means that the number was not intended to be understood simply but as something symbolic. Perhaps other commentators state this as well. I have yet to check.
1 He only refers to it as Rosh Hashanah, but I added "La'melachim" to differentiate between it and the "regular" Rosh Hashanah.