The sword in the last of the pictures you list shows a Japanese style blade with a "kagi" (hook) before the tsuba (guard). Hooks like this were not ever found on swords in Japan, I don't think.
And I wouldn't think it would even be possible to construct a blade with this as part of the same piece of metal as is shown in that picture. It would likely have to be part of the tsuba, attached separately. Either that, or the blade itself would need to be constructed using inferior methods, stamping it out from spring steel or something, and then polishing and sharpening. But I can clearly see from the picture that this is a Japanese style sword, including the curve which results from Japanese sword making processes.
Typically, a kagi would appear on Jutte (also spelled "Jitte"). Jutte were straight bars of solid iron with the kagi just above the handle. They were not swords and did not have any sharp edges. They were used a number of ways, mostly by police in Japan after swords were banned. Most people believe this was used to trap a sword and break it. But in reality, it was used to hook into clothing and pressure points.
In your video game sword, my guess is they wanted something that looked cool and could be used in the game to hook and trap either swords or chains.
I've seen hooks on the guards of other swords as well. Such as a butterfly sword popular in wing chun and hung gar kung fu. You'll see them using the butterfly sword by switching the hold from regular to reverse hold, back and forth. When switching to the reverse grip, the guard must allow it to be held that way. And that is perhaps the main reason for why the butterfly sword has that hook on it. It also adds weight to make the sword balanced. Less convincingly, it can be used to trap other swords.
Swords from Europe and the middle east might also have little or sometimes big hooks on them, usually curved with balls on the end. They're usually ornate and merely add to the appearance, not the function. They're not sharp, either. They can add weight to help balance the sword. And they can allow for a reverse grip more easily, in the case of short swords.
Just my thoughts. I'm not an expert on historical swords. Most of my knowledge is from Japanese swords.
Hope that helps.