Somebody swapped out many for "two or more" with battle, which defeats the whole purpose of the distinction.
For me fighting is an informal event (two people having a fistfight) an doesn't have to be physical, just that two or three people are involved in some type of aggressive conflict.
A battle requires that it is formal and physical fighting. It can't be a domestic spout but it can involve two people if its formal (two people in uniform involved in a melee, especially if the picture is portrayed as being part of a larger, unseen battle) but I feel large-scale, by the standards here, should be four (maybe) five or more involved in a melee.
I put "many" but someone swapped it out for "two or more", if that is the case, the definition involved two people if they are part of a larger battle makes sense but then it should be tagged as "battlefield" perhaps so its clear that its just not those two people fighting.
Hopefully this helps.
As for the posts, I don't know enough of the series but aren't they on the same side? It seems like it would be more of a fight. The second highlights the idea I put out for battle, since even though they are both in knightly-armor, they are on a battlefield so I feel that is basically a zoom-in on a larger battle. I might be wrong in using this definition since it requires what is basically a subjective decision but if we are going to use "two or more" that is really the only way to do it.
The third is a fight, Ran is basically the setting that it is occurring on since she isn't actively involved in it.
The fourth I know nothing about it beyond what I see but it seems to be mixed-martial arts in a formal arena. I'd list it as a fight because its basically involving two people fighting it out. It is formal but to me a battle is formal in that there is a battlefield or five or more people fighting it out with or without sides and to me an arena doesn't qualify as such.
Hopefully this helps.