I find it odd that many of the times people reference, or talk about the war they usually call it " the first arab- Israeli war". While it is technically not wrong, it is also not the most accurate way to describe it. In addition, a while ago I was talking to someone from the states who claimed that it wasn't really a war of independence. Can someone shed light on this?
I think it's a bit hard to nail down why terminology sometimes settles on what it does. FWIW I've heard "Israeli War of Independence" used quite frequently. But there are some reasons why "First Arab-Israeli War" is a useful name. I can't answer why one term got used more than the other, but I can provide some background on why "First Arab-Israeli War" is a useful term.
The first is who exactly the war was fought between. "War of Independence" suggests a war between a political group seeking political independence and the current ruling power. But that was a somewhat limited component of the war. The previous ruling party was the British. While there were active Jewish and Arab insurgencies during their rule, the British withdrew when the UN mandate expired without trying to maintain their rule. The most intense warfare occurred after they'd left entirely. Which makes it somewhat different from a "normal" war of independence--see the others in Wikipedia's list.
But it's also useful for contextualizing a bit. Rather than being a war where Israel declared independence, a war was fought over it, and Israel won, the conflict didn't really end permanently. Israel's independence was not accepted when the 1949 armistices were signed, and the conflicts continued with several more major wars between the same parties and largely over the same conflict. "First Arab-Israeli war" puts it in that context.
And it can also be more specific than "Israeli War of Independence". That term would presumably cover a number of conflicts--the Jewish insurgency against British rule, efforts to circumvent British caps on Jewish immigration, and conflict between Jews and Arabs within Palestine. All these were part of the Jewish campaign for independence, but did not directly involve neighboring Arab countries. This is particularly true of the conflict as the British rule of Palestine began to break down--Jewish and Arab militias were fighting openly and both were fighting the British. After the British officially pulled up stakes and Israel declared independence the conflict was between Israel and neighboring Arab countries. While the conflict continued from one phase to the next unabated, the Arab side had a significant change in who was actually fighting.