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That's what I've heard it means. So the judges don't really pluck out the other defendants' eyes due to a plaintiff complaints. However, the judge would require the defendant to compensate for the loss of eyes of the plaintiff.

A verse attract my attention

22 “If people are fighting and hit a pregnant woman and she gives
birth prematurely[e] but there is no serious injury, the offender must
be fined whatever the woman’s husband demands and the court allows. 23
But if there is serious injury, you are to take life for life, 24 eye
for eye, tooth for tooth, hand for hand, foot for foot, 25 burn for
burn, wound for wound, bruise for bruise.

Here the stories talk about paying fine (or some tort settlements it seems). I do not know if it's paid to the government or the plaintiff.

Then it talks about eye for eye, tooth for tooth. That seems to suggest that the eye for eye means fine too.

However, there is a life for life thing there. I do not expect life for life to means paying life. It seems that it's death penalty.

However, I maybe wrong.

In those verses and in other verses, is there any clear indication whether it should be fine or punitive damage or compensatory damage. Or is it just straightforward retaliation?

The Gemara on Bava Kama 83b:

אמאי (שמות כא, כד) עין תחת עין אמר רחמנא אימא עין ממש לא סלקא דעתך דתניא יכול סימא את עינו מסמא את עינו קטע את ידו מקטע את ידו שיבר את רגלו משבר את רגלו ת"ל (ויקרא כד, כא) מכה אדם ומכה בהמה מה מכה בהמה לתשלומין אף מכה אדם לתשלומין

(My translation with help from steinzaltz): Why (does it say) “An eye for an eye”? The Torah would tell you: maybe it means it literally? You wouldn’t (shouldn’t) think that, for we learned in a b’raita “(We would think) that for blinding someone, he is liable to be blinded, for breaking his arm, he is liable to have his arm broken, for breaking his leg, he is liable to have his leg broken, so the Torah tells us ״מכה אדם ומכה בהמה״ (“One who hits a person or an animal): just like one who hits an animal gets fined, so too one who hits a person gets fined.
Rashi on the verse “an eye for an eye”(sh’mos 21:24) brings this gemara as proof that he was fined:

עין תחת עין. סִמֵּא עֵין חֲבֵרוֹ נוֹתֵן לוֹ דְּמֵי עֵינוֹ כַּמָּה שֶׁפָּחֲתוּ דָּמָיו לִמְכֹּר בַּשּׁוּק, וְכֵן כֻּלָּם; וְלֹא נְטִילַת אֵבֶר מַמָּשׁ, כְּמוֹ שֶׁדָּרְשׁוּ רַבּוֹתֵינוּ בְּפֶרֶק הַחוֹבֵל (בבא קמא דף פ"ג):
“An eye for an eye”- One who blinded his friend, pays him the difference of his worth on the marketplace, the same in the other cases; but not taking the organ itself, like our teachers taught in the Gemara (Bava Kama 83).

There are many other instances in the Gemara where this is said, and many other commentators on this verse that say the same thing, but I think one example of each is sufficient.

Even before the Gemara, the Mishna on Bava Kamma 83b makes it clear :

One who injures another becomes liable for five things: damages,
pain, medical expenses, incapacitation, and mental anguish.

-Damages: If he put out his eye, cut off his arm or broke his leg, the injured person is considered as if he were a slave being sold in
the market place, and one must calculate how much he was worth before
the injury and how much he is worth after the injury; [the difference
is the damages to be paid]

-Pain: One must calculate how much a man of equal standing would require to be paid to undergo such pain.

-Medical expenses: If he has struck another, he is under obligation to pay medical expenses... If the wound was healed but reopened,
healed again but reopened, he would still be under obligation to heal
him. If, however, the wound had completely healed [even though it may
have reopened much later] he would no longer be under obligation to
heal him.

-Incapacitation: The wages lost during the period of illness must be reimbursed.

-Mental anguish: Must be calculated in accordance with the status of the offender and the offended.

Based on Rashi ad. loc., there is a debate whether נפש תחת נפש, "life for life" in that verse is to be taken literally, and the assailant/murderer is liable for capital punishment, or if that phrase, too, is figurative, like the phrases in v. 24-25, and the defendant merely liable for monetary compensation. If the latter, the defendant pays the full (market) value of the woman (as if she were to be sold as a slave) to the woman's heirs (the husband, generally). This would be similar to a modern day wrongful death suit. The debate hinges on the question of whether one who aims to kill one person but ends up killing someone else is classified as a capital murderer (and liable to be executed) or not. (See Mishna Sanhedrin 9:2; 79a)

The other clauses (in verses 24 and 25) are universally understood to mean monetary compensation, paid to the injured party, not to the government. (You expressed uncertainty on this point.) The other answers here, as well as other questions and answers on this site, speak to that discussion. Note that this is assessed as compensation, not as a fine or a penalty. This distinction has ramifications beyond the scope of this discussion.

By the way, I believe the translation you have is erroneous:

22: If people are fighting and hit a pregnant woman and she gives
birth prematurely[e] but there is no serious injury, the offender must
be fined whatever the woman’s husband demands and the court allows.

Mechon Mamre translates ויצאו ילדיה ולא יהיה אסון as "... so her fruit depart, and yet no harm follow." ArtScroll is more plain: "... and she will miscarry, but there will be no fatality [to the woman, c.f. Rashi ad. loc.]" The verse is generally understood to be discussing the rules of transferred intent as they apply to to a fetus - not a capital crime - and a pregnant woman (as a specific example of the general rule) - possibly a case of capital punishment.