There are many words in Russian for 'because'. I am unclear whether they are equivalent or are they used in different situations.
потому что: the most generic "because".
так как: "since"; more bookish and less forthright, meaning you can't use it to make an impactful "why? well I'll tell you why" statement, like you can with both
потому что and the English
(not on your list) поскольку: much the same as
так как, but you're marginally more likely to hear it in everyday life.
благодаря: "thanks to", "by virtue of", Like these English expressions, it's a preposition rather than a conjunction, so it's followed by a noun or pronoun (in the dative) rather than a clause. It can connect to a clause as
благодаря тому, что.
из-за: a reverse
благодаря if you like. Grammatically, it works the same; however, it's a "blaming" word, implying an adverse consequence and/or the cause being a fault or flaw of some sort.
(not on your list) ввиду: "on account of". Either bookish or officialese, a preposition like the previous two.
по: even more bookish/officialese than
ввиду, again with a touch of negativity and fault-finding.
ради: "for the sake of". A rare case of a non-lexical term borrowed from Old Persian (radiy, Modern Persian را -rā with a different function), which is why occasionally, you might find it in its original postpositional form (
Бога ради, etc.) but this sounds a bit obsolete.
That's an excellent question, and although Nikolay already wrote a good answer, I want to add that in order to deeply understand the exact meanings of these words, you have to look at their literal composition and use your imagination, as I explain below.
''Потому'' literally is ''по тому,'' which literally means ''on that'' (''по'' = on, ''тому'' = that). This word is thus used to introduce a logical foundation, i.e., you can literally understand ''потому'' as ''on the foundation that.'' To understand this word, imagine the foundation of a building. That's the analogy here. On that foundation, a logical construction is built. We use ''потому'' to introduce a cause, reasoning, or motivation leading to a decision or an action.
Я пошла в магазин, потому что у меня дома кончилась еда = I went to a shop because (or, literally, on the logical foundation that) no food had been left at my home.
''Так как'' literally means ''so as'' (''так'' = so, ''как'' = as) and is fully or almost fully equivalent to the English ''as'' used to connect clauses in a complex sentence. Although we often use ''так как'' interchangeably with ''потому что,'' there is a difference in the flavour: ''Так как'' rather introduces a related observation, an analogy, or a logically connected thing in order to help the conversation partner understand why something happens:
У морского черта острые зубы, так как это хищная рыба = The angler fish has sharp teeth, as it is a predatory fish.
Here, being a predatory fish is not, strictly speaking, the cause of having sharp teeth; rather, these things are just related: the fish just evolved in this way, and it is because this fish eats other fish that we classify the angler fish as a predatory fish. This is why it is more natural to use ''так как'' instead of ''потому что'' here.
''Из-за'' literally is ''из за'', which means ''from behind'' (''из'' = from, ''за'' = behind). To understand this expression, imagine the wake behind a ship. That's the analogy here. From behind a ship. That is, something happens as a result of something, usually as a side effect or local effect of something big or global happening. Here are some close English expressions: ''In the wake of,'' ''as a result of,'' ''caused by.''
Я была вынуждена закрыть свой бизнес из-за обвала рубля = I had to close my business in the wake of Russia's rouble plunge.
Here, the plunge of Russia's rouble is a global thing affecting the whole country, and my business is just something very small on the global scale. My business gets closed in the wake of a global phenomenon.
There are also many other Russian words and expressions whose meaning is close or related to ''because,'' and you need to know the exact meaning to use them properly. You listed only a small fraction of them. Here are some words and expressions that instantly come to my mind:
для того, чтобы
с того, что
почто (e.g., ''Почто боярыню обидел, смерд?'')
да (in conversations like: ''Почему ты это сделал?'' ''Да надоело терпеть просто.'')
в связи с тем, что
с учетом того, что
на основании того, что
The exact meaning is usually easily seen from the literal composition, as I explained.