I read about Nathan Hanavi and Gad Hachozeh here.
Are those the same job?
The Targum Yonason on Shmuel 9:9 seems to equate the term Chozeh with Roeh.
(Source provided from Sefaria.org)
אומר כי יש הבדל בין הנביא בדור אחרון ובין הרואה לפנים, שהנביא נקרא שמו על שם דברותיו ותוכחותיו (מלשון ניב שפתים) על כי ה' דבר בו וישלחהו להגיד לעם פשעם ולבית יעקב חטאתם, ולא שלחהו במלאכות ה' רק לצורך עניני הכלל והגוי כולו ולא חל רוח הקודש עליו להראותו דברים פרטיים הנוגעים אל היחידים, אבל הרואה נקרא בשמו על שם שהיה צופה ברוח הקודש השורה עליו כל הדברים ההויים והעתידים וגם עניני היחידים ופרטיהם, ועל כן היו בורחים מן הנביא אשר היה תמיד מגיד פשע ומוכיח עלי עון, ורדפום אחר הרואה באשר כל אחד השיג ממנו מבוקשו על הגניבה ועל האבידה ועל החולה ועל האשה והבנים, והיו נותנים לו תשורה ומתן בעבור שהתבודד לעיין על עניניהם,
(The verse is) saying that there is a difference between a Navi in the later generation and the Roeh of old, for the Navi is called a "Navi" because of his words and rebukes (from the term "uttering of the lips) for G-d spoke to him and sent him to tell the people their sins and the house of Jacob what they were guilty of, and was not sent with the the work of G-d, rather for the needs of the general population and the nation as a whole. Ruach HaKodesh did not rest upon (the Navi) to show him specific matters pertaining to individuals. However a "Roeh" is called such because he would see with Ruach HaKodesh and matters of the future as well as matters of individuals and their details would rest on him. Therefore (the people) would run from the Navi who would always tell them of their sin and rebuke their transgression, and pursue the Roeh in that everyone would inquire of his and ask regarding their stolen and lost items, about their maladies and about their wives and children and would give a offering and gift because of because he (the Roeh) would set himself to look into these matters
Hope this is helpful
HALOT suggests this etymology for נָבִיא:
to be associated with the Akkadian verb nabû to name, call; from which is derived נָבִיא, which may have an active sense “speaker, herald, preacher” or (more probably) a passive sense “one who has been called”
While it notes that חֹזֶה is the participle of חזה, which has the meaning of "to see, behold". Some of the examples it gives are:
God sees Ps 11:4 17:2 Sir 15:18, is watching over with עַל cj. Jb 3429 (Ehrlich);
man sees God Ex 24:11 Jb 19:26f חָזָה מַחֲזֶה to see a vision Nu 24:4.16 Ezk 13:7
to see as a seer Am 1:1
חָזָה לוֹ to select for oneself Ex 18:21 ח׳ בְ to see with pleasure, satisfaction (→ ראה 10) Mi 4:11 Ps 27:4 (= experience) Jb 36:25 Song 7:1
The connection between the two words is obvious: prophets and seers were (or falsely claimed to have been) given visions of the future by God.
But actually this is only a small part of the role of a prophet. Visions of the far future are rare in the scriptures. Instead most of their messages from God implored the people to repent of their sin and turn back to him in faith. When the prophets foretold judgement it wasn't necessarily because they were given a vision of the future, but because Israel's covenant with YHWH promised curses on the sinful (Deuteronomy 27-28). They didn't need visions to know that God would judge the unrepentant. They were called prophets because they obeyed God's call to them to speak boldly to the nation.
But although the term "seer" may more strongly imply that the person has been given visions than a "prophet" implies (who might receive only a message rather than a vision), it seems to me that the two terms are pretty synonymous in the scriptures.
Particular people are consistently called one title, such as Nathan the prophet and Gad the seer, even though their roles were very similar. Gad's speech in 2 Samuel 24:11-13/1 Chron. 21:9-12 did not involve a vision, and neither did Jehu the son of Hanani the seer in 2 Chronicles 19:2-3.
In Amos 7:12 the seer Amos is instructed to prophesy.
חֹזֶה is very commonly used in parallel with other terms without clear differences:
In Chronicles, after it has finished telling a king's story, it often refers the reader to go to the writings of both a prophet and a seer:
Aside from these parallels, the words are largely complementary - most authors only use one of the words.
The term Navi is a generic term for one to whom G-d speaks. It is taken from the expression ניב שפתים (Isaiah 57:19) which is the idea of the lips drooling. When a Prophet on a level lower than Moshe experiences prophecy, they lose control of many of their bodily functions including drooling. The expression Navuah is the process of communication.
Chozeh is referring to a prophet who receives a specific level of prophecy on the level of vision (See Zohar, parshat Pekudai, 248:a, chapter 533-536 for details). This is also found at the end of tikkun 39 in Tikkunei Zohar, 79b.
ועוד ויאמר אלקים יהי אור. דא נבואה במראה דאתמר בה (במדבר יב) במראה אליו אתודע ואיהי חזון
That is the level of Chazone, like Isaiah 1:1. This expression is also used in connection with the level of Moshe Rabbeinu's prophecy like is found in Tehillim 58:11.
יִשְׂמַח צַדִּיק כִּי חָזָה נָקָם פְּעָמָיו יִרְחַץ בְּדַם הָרָשָׁע:
In this example, Moshe's prophecy is expressed with the phrase of chazah, like in a vision. In chapter 533 from the Zohar cited, this is described by the phrase Zeh.
מאתר דא, ינקי כל אינון מאריהון דחכמתא, דקיימן למנדע במראה, או ברזא דחלמא
This phrase is described as being able to look at something clearly and say, This is the thing. A good discussion of the concept of Moshe's level of prophecy can be found here.
The lower levels of prophecy come from the level of hearing and are called Bat Kol and Ruach HaKodesh among many other names. In contrast to the level of Zeh, this lower level is described by the phrase Koh (כֹּה אָמַר ה')*
This is discussed in the commentary of Rabbeinu Bechai on the Torah (Devarim 33:8) and in Sha'ar Ruach HaKodesh (Drush 1) of Rabbi Chaim Vital.
The different terminology is only distinguishing between the level from which the prophet receives, meaning the source of their prophecy or the level of his reception, meaning the prophets ability to contain and absorb the message. In general, prophets serve the purpose of helping the people to return to the service of G-d. That may be directed to the Jewish people or to the nations at large.