To my knowledge, it was actually the allied forces that coined the term "Banzai", as in a "Banzai charge" made by the Japanese fighting forces. The word comes from the full Japanese term "Tennōheika Banzai" ("Long live the Emperor"), and referred to what was essentially seen as a suicide rush charge against the enemy by Japanese soldiers who, when facing what was believed was all but certain defeat, felt that just rushing the enemy and going into a scenario where near-certain death was on the table (think of the idea of "suicide by cop" as a somewhat appropriate analogy I think) was a most honorable way to meet one's death rather than the dishonorable act of surrender. Adopting this method of death was known as gyokusai, or "shattered jewel".

The Japanese during WWII (as well as prior to and even since) were big-time indoctrinated into the concept of meeting, what in their eyes at least, was the most "honorable" death possible at either the hands of the enemy when it was felt defeat was otherwise clear, or alternatively, via self-inflicted ritual suicide self-stabbing known as Seppuku (also called Harakiri in English). This all followed portions of the Bushido honor code that the Japanese were (in)famous for following in combat.

Occasionally, Banzai charges did prove to be successful, even though they typically resulted in the loss of many men. Banzai charges were used by the Japanese against Chinese forces in the 1930s for example to some effect, which made it an apparent last-ditch but viable military tactic in their eyes, as opposed to unconditional surrender.

As for whether the Japanese commanders actually shouted out "Banzai" or "Tennōheika Banzai" prior to the charge, using it as a rallying cry, that is really not well documented to prove one way or the other, though it is presumed the Japanese commanders did say/yell something to that effect as the charge began to stir the troops and rally them to make the charge. Again though, it seems it was the English-speaking allied forces who actually coined the basic term "Banzai charge" to describe this suicidal rush from the Japanese soldiers.

I don't know if there are any pictures or film footage of an actual Banzai charge shot as it was taking place (I have never seen any that can be positively ID'ed as being "real" imagery or footage of these crazy events actually taking place), but there certainly is documented footage from the Pacific Theater in WWII of the immediate aftermath of these style of attacks available in the form of post-combat forensic footage of the battlefield shot by US cameramen. Here for example is a brief film clip taken after a famous and very massive (the largest of WWII) Japanese Tennōheika Banzai charge that occurred on the island of Saipan in the Mariana Island chain in 1944 against American soldiers. This attack was (barely) repelled by US forces, resulting in the death of about 4300 Japanese soldiers and that almost succeeded in wiping out the 1st and 2nd Battalions of the 105th U.S. Infantry Division, who suffered over 650 fatal casualties during this failed Japanese suicidal rush attack.