It's certainly true that the Arab states do not refer to it as the Six-Day War, given that it would bring up the embarrassment they felt following its conclusion. They tend to refer to it as the "67 war" or the "June War", as you've noted. The decision to name the war was made following the war's end, as Israel was basking in the glory of it. Yitzhak Rabin was Chief of Staff of the IDF during this time, and much of the praise was heaped upon him as being the "architect" of the war's success. For reference, the IDF Chief of Staff is the supreme commander of the IDF, and is nothing like a Chief of Staff in the US White House, for example.

Rabin was elevated to "near-icon" status, according to Michael Oren in Six Days of War, and Rabin was given the opportunity to actually name the war in an interview not long after the war ended. Names proposed included "The War of Daring", "The War of Salvation", "The War of the Sons of Light", etc., but he chose "The Six Day War", choosing to invoke the story of Creation (wherein creation took 6 days, and on the seventh there was rest, in the biblical story). The name served another purpose to him, however. As he stated to Israel's army magazine on July 5, 1967 (quoted from Rabin: 20 Years After by Robert Slater):

I do not think this war needs a special name. Its extent and results speak for themselves. All the names suggested until now...are too pretentious. The simplest, and to my mind, the most apt, is the Six Day War, reflecting the six days of creation.

Rabin's name easily stuck, given his stature and importance and dismissal of the other names. Part of his goal with the simpler, less "ostentatious" name was to try and tamp down the elation of Israelis, to remind them that despite winning the war, they could not have false illusions about their situation (they were still surrounded by enemies) and that they should not glorify the war overmuch. Rabin, when he received an honorary doctorate following the war, said himself that the war was one of "salvation", but also noted that "[the soldiers] cannot rejoice wholeheartedly. Their triumph is marred by grief and shock...". He was keen to reflect on the suffering the war brought, rather than just the victory, which was to be expected from the IDF's Chief of Staff and also from Rabin's personality.