It appears that Jewish kings wore crowns. See for example Shmuel Bet 1:10, 12:30.
The Torah does not instruct a king to wear a crown. Obviously the king can wear pretty much whatever he wants within reason but given that wearing a crown was a custom of non-Jewish kings and the lack of a mitzvah to wear one I would like to know if there was any particular religious reason that Jewish kings chose to wear a crown.
One could suggest that wearing a crown for a Jewish king is optional.
The Mishna in Sanhedrin 22a lists laws about a king's possessions that one may not use:
מתני׳ אין רוכבין על סוסו ואין יושבין על כסאו ואין משתמשין בשרביטו ואין רואין אותו כשהוא מסתפר
One may not ride on the king’s horse, and one may not sit on his throne, and one may not use his scepter, and one may not see him when he is having his hair cut,
Notice how "crown" is not mentioned in this list.
However:
The Rambam in Hilchos Melachim 2:1 basically quotes this Mishnah verbatim BUT adds "crown" to the list!
אֵין רוֹכְבִין עַל סוּסוֹ. וְאֵין יוֹשְׁבִין עַל כִּסְאוֹ. וְאֵין מִשְׁתַּמְּשִׁין בְּשַׁרְבִיטוֹ. וְלֹא בְּכִתְרוֹ. וְלֹא בְּאֶחָד מִכָּל כְּלֵי תַּשְׁמִישָׁיו.
The king must receive great honor. He is to be venerated and feared by everyone, as it says, “Surely, shall you place” (Deut. 17:15) – that is, his awe is to be upon you. No one may ride his horse, sit on his chair, use his scepter or his crown or any of his serving vessels.
Here we see the Rambam adds "crown" to this list.
Q: Why and how could the Rambam add to the list of items listed in the Mishna?
A: One could suggest that while a Jewish king may have a "crown," they're not obligated to wear it.
TL:DR if wearing a crown was optional there must be no religious reason to wear one.
It's an interesting question, and I suspect you won't find a clear source. Even a surface reading of the Tanach makes it clear that there were many customs which the Jews either had from ancient times, or had adopted from their surroundings. The Tanach isn't a history book and doesn't detail things that don't have importance for all time.
Nonetheless, it's interesting to examine the various terms for "crown". The first you cited is נזר, which we see that Shaul wore. It appears in the context of the High Priest, who also wore a נזר הקודש - the golden tzitz, a metaphorical crown - on top of his hat. (And this may be why the rabbis refer to the three crowns: one of kingship, one of priesthood [see Kiddushin 66a], and one of Torah [possibly a reference to the crownlike border called the zer around the aron, ark, but also to the respect owed to a Torah scholar].)
The second source you cite is of the word עטרת. Here we only see this in the context of David taking it from the head of an Ammonite king and putting it on his own head, a clear sign of transferal of power.
The other Hebrew word for crown (common in modern Hebrew) is כתר. Interestingly, the only king who has this in Tanach is Ahasuerus in the book of Esther, but this word appears a lot in the Talmud.
The Rambam (Hil. Melachim 2:5) states about the king ומשים כתר בראשו, he places a crown on his head. So, contrary to the OP's assumption, it would appear that the Torah does instruct him to do so. (Now, it could be that the Rambam means it as an optional thing, but if so, why state it at all? A king presumably also has the option of choosing what color his royal robes will be, for example, but there's no need to codify that.)
In fact, the Rogatchover (Tzofnas Paaneach on Bereishis 42:8) references in this connection a Gemara (Chagigah 16a) that says that anyone who gazes at a king (nassi), his eyes grow dim, and so suggests that this is precisely why he should wear a crown - so that they don't gaze at him. (Though I don't really understand that point - wouldn't that draw people's attention all the more? - unless he means that this way people will focus on the crown rather than on his face.)
In the book of Zechariah 6:10-14 God instructs that crowns be made for the high priest and the ruler 'Tzemach'. Though this ruler is not officially a king (which would have been viewed as rebellion against the ruling Persian kingdom) most commentaries identify him with Zerubavel who descended from the family of King David amd may have had a quasi-kingly standing among the people.
Thus, though there may have not been a specific commandment for a king to wear a crown, it seems religiously appropriate.
