So far I understand that a long exact sequence refers to a countably infinite exact sequence, and a short exact sequence is a special type of finite exact sequence, beginning and ending with $0$ and containing precisely three nonzero objects in between.
However, I don't understand how the definition of an exact sequence requires finite exact sequences to begin and end with zero. The condition only seems to describe each arrow individually, so seemingly given any finite exact sequence one should be able to remove the leftmost and/or the rightmost spaces as well as their map into/in from the other spaces, and still have an exact sequence.
For example, given a short exact sequence, don't the central three (nonzero) objects and the two arrows between them still form an exact sequence by themselves? So what is the motivation for always including and considering the map from $0$ on the left and the map into $0$ on the right? It seems like redundant information, in addition to being visually unappealing.
And whenever I see examples of finite exact sequences, even not short ones, they always seem to begin and end with $0$ for some reason. What is the motivation for any of this?
I don't know how to use Tikz, so if any of what I am saying or asking is unclear, I will draw, scan, and post example diagrams.