Wouldn’t the English word Crusader have the same meaning as mujāhid?
The word “Crusader” comes to us from Latin, and is derived ultimately from the Latin word “crux”, meaning “cross” as in the symbol of Christianity. This is because the “Crusades” were called the “wars of the Cross”, fought in behalf of and for Christ’s endangered people in the East.
The Arabic equivalent, which is used quite commonly, is (singular: صليبي, Salībī; nominative plural: صليبيون, Salībiyyūn; oblique plural: صليبيين, Salībiyyīn) — derived from (صليب, Salīb, “cross”) — and meaning “of or pertaining to the (Christian) cross”. In other words, Cross-ites or Cross-ians, as it were. Which is quite close to “Crusader” on a variety of levels.
Both words refer, originally, to one or to those who fight(s) for Christianity.
Now, the root for (مجاهد, mujāhid) is (جهد, jahada), meaning (to) “struggle”. Now, the common active form would be (مجهد, mujahid), but (جهد, jahad) is different from the IIIrd form thereof, (جهاد, jihād), and the IIIrd form’s corresponding active form is (مجاهد, mujāhid). Thus, (مجاهد, mujāhid) does not mean “one who struggles” but rather “one who wages religious war”.
(Sidenote: So, note this: the fact that (جهاد, jihād, “to wage religious war”) is derived from (جهد, jahad, “to struggle”) means nothing. In fact, so central is this concept that the root has a special form that means specifically “religious war”. And so if someone is taking about (جهاد, jihād) and not (جهد, jahad), one is talking solely about religious war. None of this spiritual stuff that is taken for the normative interpretation.)
As such, “Crusader” and “mujāhid” mean different things semantically and etymologically. In a sense, it is possible to call Crusaders mujāhidun but, because of the word’s origin in “cross” one cannot call a Muslim of any sort a Crusader. It is, properly, an exclusively Christian term. (Similarly, one cannot or ought not to apply the word “Crusader” to any non-Christian.) It’s a logical issue: why would a non-Christian fight for the Cross?
Now, their connotation in the native languages is similar: both refer to holy warriors; both words have a positive and, indeed, even reverential connotation. However, when Christians use “mujāhid“, it is in a derogatory sense, just as when Muslims use “Crusader”. And so, as far as words are concerned, it’s still a mess.
In other words: “mujāhid” (“one who wages religious war”) does not mean the same thing as “Crusader” (“one who fights for the cross”) — at least explicitly. And while Crusaders can be said to be mujāhidūn of sorts, not all mujāhidūn are Crusaders or can be called such. However, there is something to be said about connotation.