Here are the words that roughly mean "first" that I'm aware of, but feel free to add more:
What are the differences (if any) between all of them?
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Ultimately because Japanese does not have the cardinal/ordinal dichotomy built in the grammar. Most of what you list are counters addressing different notions that may translate into a single word "first".
最初: at the start; first in (chrono)logical order; firstly
一番: literally "number one", "1 in number"
It is a very basic word, but not always ordinal (simply referring to a numeral ID). When used ordinal-like, it roughly means "first in significance", which is why it also has the meaning "most" for adjectives. Translation-wise it does not correspond well to "first", but usually "best", "worst", "biggest" etc. according to the context.
一番目: "number one in order"
～目 is a suffix to any counter to embed ordinality. Thus it is but a special case of context-aware
n + [counter] + 目 construction e.g. 一個目のリンゴ "first apple", 20回目の誕生日 "20th birthday". The specific 一番目 is only appropriate for uniform and conceptual items, such as the first element of array by index.
一次/第一次: first stage/phase; first in existence
In my opinion, it should be understood as "primary" most times, in the way it is the most essential or the origin, then 二次 "secondary", 三次 "tertiary".... (Adding 第 or not is discussed below.)
一位/第一位: the first place; first in rank
As you can imagine, this is widely used in competitive situations where participants ranked by number. But in races competing on speed, 一着 "first arrival" may be preferred.
首位: top (= the first place)
第一: literally "order one", "priority one"
This is etymologically a Chinese phrase (duh) rather than a word that has a monolithic meaning in Japanese, but roughly speaking, "before any(one/thing)" as a standalone word, or "the first of a fixed set of items" as prefix.
第- or not: This is mostly a remnant of Chinese grammar, where 第 is required for ordinals. However, it sometimes has actual meaning when the counter can be semantically cardinal or ordinal. 第- and -目 are slightly different, too, that the former suitable for items inherently serial or sortable, and the latter those not.
二回目 "the second time"
第二回 "the second (round of some event)"