Why call on an onto function surjective?

Question

Why call on an onto function surjective?

Answer 1

Why use the word "surjective" which rhymes with injective?

Your suggested alternatives, right-totality and left-totality, also rhyme, and furthermore understanding them depends on the order of writing things, which is not canonical. (There's no reason why you shouldn't have $Y \leftarrow X : f$, and in fact when writing about category theory you see notations like this pop up now and then. Also note that there are languages written right to left in general.) If you speak a language with influence from romance languages you might recognize "sur" as meaning something like "onto", which provides easy intuition. In the end it doesn't matter too much which words we use; it is easy to get used to these concepts.

Also while on the topic why are functions commonly defined as triplets?

This is actually not, typically, how functions are defined in ZFC (which is often the implied background of mathematics). We sometimes use the triple of the definition specifically because we want to associate the codomain to the function; the statement that "a function is surjective" then makes sense without explicating a codomain, and it also has real implications (like the existence of a section).

If one wants to specify that the image of their function is a subset of some other set, why can't they just write $\mathrm{img}(f)\subseteq Y$?

Because it's more work.

Why purposefully restrict the expressive power of ones notation by defining a notion of codomain and requiring composition adhere to rules formulated by it

Because it's more convenient in practice. In practice, it's more convenient to keep the functional notation $f: X \to Y$ even when $f$ does not surject onto $Y$, and when you have a function $g: Y' \to Z$ with $Y' \subseteq \mathrm{img}(f)$ you just write $g \circ f$ for the intended composition, even if it is not strictly speaking correct.

Ultimately this is the answer to all your questions: it doesn't really matter that much. All notation has advantages and disadvantages. Mathematicians have settled on what they have settled on, often with decent reasons. Your suggestions would provide easier notation only in marginal situations, and make things worse slightly more often than that. High-level serious proofs would rarely, if ever, be affected.