Let me say right away, I am very fascinated by the topic of Linguistics, Lexicology and Etymology, from a young age I was interested in it, so I often made my own analyses on different topics through this Methodology. So I will give you an answer within that style. You might find it interesting. I did my research on this topic a few years ago and came up with some thoughts.
I can tell you that the words «Occult» and «Occultism» definitely share a common origin from a Latin root and are indeed related to the meaning of «Secret», «Hidden».
The base form in Latin is considered to be the adjective «Occultus» which is derived from the verb «Occulere» with the meaning «To Hide», «To Conceal». «Occultus» translates as «Hidden», «Secret», and the verb «Occulere» is formed from the prefix «ob-» and the root «-celere» (from Latin «Celare» — «To Hide»). The Morphemic composition of «Occulere» literally can be interpreted as «To Close (in front of Something)», «To Make Invisible». «Occultus — Occulta — Occultum» in Latin act as adjectives, and the noun «Occultum» can mean «Mystery», «Hidden», «Mysterious».
Latin «Occultus» gradually passed into French as the adjective «Occulte». From here, the spelling with two «cc», reflecting the Latin base, became entrenched in many European Languages. English borrowed the word «Occult» through the French «Occulte».
In English, «Occult» was formed both as an adjective («Occult Science») and as a noun («The Occult» — «Occultism» or «The Field of Secret Knowledge»).
German inherited a similar pattern: «Okkultismus», «Okkult». German uses a doubled «k» instead of the Latin «cc», which is characteristic of the transfer of Orthography in German.
Similarly, Italian has «Occulto», Spanish has «Oculto», etc. In general, Romance Languages are quite close to the Latin form, while Germanic Languages adjust the spelling to their own Phonetic-Graphic Structure. For example, doubling the consonant «k» instead of «c», or retaining the «c», which in Germanic reads «/k/». I find such details very interesting.
Unfortunately, I have not found a direct equivalent of «Occultus» as a calque in Ancient Greek, but there is a related word «κρυπτός» («kryptós» — «Hidden») formed from the root «κρυπ-» («kryp-»), which, as far as I understand, goes back to the words «Hide», «Cover». The term «Occult» is not historically derived from Greek, but in the Vocabulary associated with Esotericism and Secret Knowledge, parallels are often drawn between «Occultus» (Latin) and «κρυπτός» (Greek) due to the proximity of the concepts of «Hidden», «Secret».
For comparison, in Old Germanic Languages (Gothic, Old High German, etc.) there are verbs with the meaning of «Hide», «Cover», but I have not found a systematic borrowing from Latin «Occultus» or a direct parallel. An example may be Old English «Helan» («To Hide», «To Cover») — homonym with modern «To Hide», but this is another Etymological Line, not directly related to «Occultus».
The original meaning of «Occultus» in Latin is «Hidden», «Concealed». Over time, within the Occult Sciences, the meaning has expanded: it is not just about physical concealment, but Metaphysical, Sacred, connected to Secret Knowledge, as I noted earlier.
In the European Traditions of the Renaissance Era and later, the word «Occult» found its way into texts on Alchemy, Astrology, and Magic, gradually acquiring a close connection with Esotericism and Mysticism. The concept of «Occult Philosophy» and then «Occult Sciences» emerges.
It turns out that what the different Languages have in common when considering the word «Occultic», «Occult», «Occulte» and etc., this the idea of the Hidden, the Mysterious, the Secret. It seems that no matter what Phonetic and Orthographic changes occurred in the borrowings, the Semantic Core remained unchanged.
Lexical parallels are also found in other Language Families. For example, «Secret» in Slavic Languages, «Hidden» in Germanic Languages. Although they have a different Etymological Basis, but functionally these words are close to the Latin «Occultus».
I was a little disappointed when I found no direct borrowing of the Latin word «Occultus» (and words formed from it) in Ancient Egyptian, Sumerian and Sanskrit languages. However, each of the mentioned languages certainly has its own words and expressions that convey the Idea of «Secret», «Hidden» or «Mystical», being in a sense analogs.
Obviously, the Ancient Egyptian language did not borrow the Latin «Occultus». But it has a whole stratum of Vocabulary connected with the ideas of the word. The word «Sheta» («štꜣ», «št3») means just about «Secret», «Hidden». As far as I understand, the word is found in texts, in the context of Sacred Knowledge to which the Priests had access.
The concept of «Heka» is often translated as «Magic» or «Magic Power» and indicates that the Ancient Egyptians meant it literally as Witchcraft, although they did not call it «Occult» in the Latin sense.
If I understood the translation correctly then, the name of the God Amun (Amun, «imn» in Egyptian transliteration) translates as «Hidden» «Unveiled». His Image is described by the Egyptians as the Force, «Bearing the Mystery» or «Hidden».
In Sumerian language there are words and phrases expressing the concepts of Hidden, Secret or Magical Knowledge, especially within the framework of Rituals, Spells. For example, there are known terms related to «Magical Word» or «Spell», one of the designations is «nam-šub», which can be interpreted as «Magical Formula», «Incantation». This term describes the Verbal Influence on Reality, in essence the Hidden, Sacral Power of the Word. Obviously, we are talking about Mantras, and more specifically, the very essence of the Mantric Formula, about Necronomicon. I found out that the Ancient Sumerians conducted entire Ceremonies of Opening the Chakras and Meditations on the Chakras. In the context of the meaning of the word, this makes sense.
Sanskrit has its own terms that convey the idea of «Hidden», «Arcane», «Secret» and «Mystical». For example, «guhya» («गुह्य») — «Secret», «Covert», «Hidden» (from the root «guh» — «To Hide»). This word is often found in Writings of Religious and Philosophical nature, for example I have found it in Vedic and Puranic Literature. There is also the word «rahasya» («रहस्य») — «Mystery», «Secret», which also carries the meaning of something requiring Initiation or special Knowledge. When speaking of the «Secret» or «Hidden» aspects of practices like Meditations, Spells, Rituals, there are various compound words in Sanskrit with «guhya-» components or other terms related to Esoteric Knowledge.
In general, the word itself, based on my analysis, belongs to the Latin-European Lexicon, I could not find more distant roots. Nevertheless in very many Ancient Cultures there are their own terms and Concepts conveying the Idea of «Mystery», «Hidden Knowledge», «Initiation», «Magic» or «Spiritual Practices», in one form or another, but apparently the meaning remains the same.
Because, obviously, it is also connected with the Right Hemispheric Side, with the Yin Aspect and Psychic Abilities, because «Darkness» is something «Undisclosed», «Hidden», but which can be accessed, to unlock the Potential of this Power. But also here there is a connection with the Soul, Akasha, «Lucifer's Light», which is not natural Light, but «Etheric», we can say, that is why it is «Secret», «Dark», «Mystical», because at the same time it is also Enlightening, Enfolding.
Of course, I can be mistaken in my analysis from the viewpoint of Linguistics, because I am human and not Ascended, but this is my personal analysis, on which I spent a lot of time long ago to re-read a lot of various information, and still it may contain some omissions. I hope it was interesting for you to read all this, I wanted to give as deep answer as possible.