tefilla - Explain differences in terms used for "prayer"

Please explain the differences in the types of "prayer" that is meant when these terms for "prayer" as found in Tanac"h are used. I have intentionally omitted English translation for these verses, as for purposes of this question, I am not just seeking literal meaning. I am looking more for the difference in nuance in the term and whether a specific type or method of prayer is implied, esp. in the verses cited. Why was that term used over the other? Also, do any of these terms have practical halachot in terms of the way we should currently pray?

עתר

Genesis 25:21:

וַיֶּעְתַּ֨ר יִצְחָ֤ק לַֽיהוָה֙ לְנֹ֣כַח אִשְׁתּ֔וֹ כִּ֥י עֲקָרָ֖ה** הִ֑וא וַיֵּעָ֤תֶר לוֹ֙ יְהוָ֔ה וַתַּ֖הַר רִבְקָ֥ה אִשְׁתּֽוֹ׃

Exodus 8:5:

וַיֹּ֣אמֶר מֹשֶׁ֣ה לְפַרְעֹה֮ הִתְפָּאֵ֣ר עָלַי֒ לְמָתַ֣י ׀ אַעְתִּ֣יר לְךָ֗ וְלַעֲבָדֶ֙יךָ֙ וּֽלְעַמְּךָ֔ לְהַכְרִית֙ הַֽצֲפַרְדְּעִ֔ים מִמְּךָ֖ וּמִבָּתֶּ֑יךָ רַ֥ק בַּיְאֹ֖ר תִּשָּׁאַֽרְנָה׃

צעק

Exodus 8:8:

וַיֵּצֵ֥א מֹשֶׁ֛ה וְאַהֲרֹ֖ן מֵעִ֣ם פַּרְעֹ֑ה וַיִּצְעַ֤ק מֹשֶׁה֙ אֶל־יְהוָ֔ה עַל־דְּבַ֥ר הַֽצְפַרְדְּעִ֖ים אֲשֶׁר־שָׂ֥ם לְפַרְעֹֽה׃

Exodus 14:10:

וּפַרְעֹ֖ה הִקְרִ֑יב וַיִּשְׂאוּ֩ בְנֵֽי־יִשְׂרָאֵ֨ל אֶת־עֵינֵיהֶ֜ם וְהִנֵּ֥ה מִצְרַ֣יִם ׀ נֹסֵ֣עַ אַחֲרֵיהֶ֗ם וַיִּֽירְאוּ֙ מְאֹ֔ד וַיִּצְעֲק֥וּ בְנֵֽי־יִשְׂרָאֵ֖ל אֶל־יְהוָֽה׃

פלל

Numbers 21:7:

וַיָּבֹא֩ הָעָ֨ם אֶל־מֹשֶׁ֜ה וַיֹּאמְר֣וּ חָטָ֗אנוּ כִּֽי־דִבַּ֤רְנוּ בַֽיהוָה֙ וָבָ֔ךְ הִתְפַּלֵּל֙ אֶל־יְהוָ֔ה וְיָסֵ֥ר מֵעָלֵ֖ינוּ אֶת־הַנָּחָ֑שׁ וַיִּתְפַּלֵּ֥ל מֹשֶׁ֖ה בְּעַ֥ד הָעָֽם׃

Deuteronomy 9:20:

וּֽבְאַהֲרֹ֗ן הִתְאַנַּ֧ף יְהוָ֛ה מְאֹ֖ד לְהַשְׁמִיד֑וֹ וָֽאֶתְפַּלֵּ֛ל גַּם־בְּעַ֥ד אַהֲרֹ֖ן בָּעֵ֥ת הַהִֽוא

חנן

Deuteronomy 3:23:

וָאֶתְחַנַּ֖ן אֶל־יְהוָ֑ה בָּעֵ֥ת הַהִ֖וא לֵאמֹֽר׃

I Kings 8:30:

וְשָׁ֨מַעְתָּ֜ אֶל־תְּחִנַּ֤ת עַבְדְּךָ֙ וְעַמְּךָ֣ יִשְׂרָאֵ֔ל אֲשֶׁ֥ר יִֽתְפַּֽלְל֖וּ אֶל־הַמָּק֣וֹם הַזֶּ֑ה וְ֠אַתָּה תִּשְׁמַ֞ע אֶל־מְק֤וֹם שִׁבְתְּךָ֙ אֶל־הַשָּׁמַ֔יִם וְשָׁמַעְתָּ֖ וְסָלָֽחְתָּ׃

(The above verse also has a form of פלל)

Answer

First of all, the midrash tells us that there are many more verbs that express prayer in the bible (see first comment here for a list of six sources that list at least ten different names for prayer). There are also many interpretations regarding the differences between them.

There isn't one commentator that I know of who lists all of them and compares them (this would be an interesting project to undertake if one has some free time). However here's a link that starts you off with a comparison between the ten main ones. And here's another one stating that the strongest prayer is expressed by "צעקה".

"עתירה" isn't listed in the above comparison, so I'll add here that both Rashi and Malbim (at least) explain that anytime this word is mentioned the meaning is: praying a lot. Rashi brings proof for this explanation from several sources.

This question touches upon the general subject of milim nirdafot (synonyms). Some commentators, such as Radak, Ibn Ezra, Ralbag, and more, say that the bible expresses itself many times twice, as a literary tool. To them, two different words can mean the exact same thing, and the Torah (or other books) uses them interchangeably for no meaningful reason.
Others, like the Malbim and the Vilna Gaon, said that there are never two words that are used in the bible and that mean the exact same thing. There's always a meaningful difference between them, and a reason for using both (or more).
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