In this context the first domain $\theta_1\left(f\right)$ and second domain $\theta_2\left(f\right)$ represent the set of pre-image elements, and the set of image elements, AKA range.
The following is from BBFSK, Part A, Section 8.4:
An important class of relations consists of the functions, defined by the requirement of uniqueness $\forall_{x}\forall_{y}\forall_{z}\left(\left(xry\land xrz\right)\implies y=z\right).$ [$\dots$] The function $f$ is a mapping of the first domain $\theta_1\left(f\right)$ onto the second domain $\theta_2\left(f\right)$: if $\theta_2\left(f\right)$ is contained in a set $\mathcal{A},$ we say that $f$ is a mapping into $\mathcal{A}.$
Apparently that is where they introduce the term mapping, and by the emphasis using italics, I assume it is intended to be a definition. Is it correct to understand this as: the term mapping means a correspondence between two sets, or between a set and itself such that the set of image elements is the second domain $\theta_2\left(f\right)$ of a function $f$. Specifically, for every argument (pre-image) element there is exactly one image element (definition of function). In other words, all mappings are single-valued.
Furthermore, this distinguishes between the term mapping and function in that a mapping has a codomain which is not ncessarily covered by image elements, whereas a function ncessarily covers its second domain.
I am particularly interested in this question as it pertains to computer science, and such fields as relational database schema and UML. I used to think that there was such a thing as a many-to-many mapping. Apparently the use of the term relation regarding many-to-many correspondences is consistent with mathematical usage, but the term mapping should be restricted to many-to-one relations, where many may be one.
Is this correct?
There term mapping is just describing the verb of what the function does to its domain onto its codomain. That is, the function maps the first domain onto the second domain, since it is of course surjective onto its image, and if the second domain is contained in another set, we say it maps into that set, even though it is not surjective.
So, a mapping and a function are the same things. Noone defines a function as a being surjective onto its second domain.
As a last note, noone calls these terms first domain and second domain. If we have a function
$f: A \rightarrow B$
we say that $A$ is the domain and $B$ is the codomain.
Regarding many-to-many relations, this is exactly just a relation, and not nessessarily a function, since a function only has one output per input. There is the notion of multi-valued functions, but I wouldn't worry about that for now.