I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way.
Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions ...
Despite the names of these fields, as a student I found real analysis more
abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples:
in abstract algebra: The order of a subgroup $H$ of a finite group
$G$ divides the order of $...
Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right?
When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year. The introduction to abstract algebra used "Numbers, Groups, and Codes", ...
In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.
I personally use the following terminology:
A relation $R \subset A \times B$ is said to be single-valued if $(a,b_1) \in R$ and $(a,b_2) \in R$ implies $b_1 = b_2$.
A relation $R \subset A \times B$ is said to be total if for all $a \in A$ there exists $b$ in $B$ with $(a,b) \in R$.
A relation which is both single valued and total is a function.
Search [mathematics.se] using the (reference-request) tag, and the (abstract-algebra) tag. There have been a number of posts, from students at various levels of study in abstract algebra, seeking text recommendations. One such post, mentions Lang's Algebra. It's a very thorough text (scan the Table of Contents here, but doesn't cover category theory ...
I taught myself the basic material from the first 6 chapters of the 2nd edition of Rotman's Advanced Modern Algebra, and I loved it and felt things were explained clearly. Earlier I had tried teaching myself out of Artin's Algebra but got stuck or spent too much time trying to understand with details of the wallpaper group (which in retrospect I wish I hadn'...
Permit me to endorse @halfbloodprince's recommendation.
Stillwell's four pillars are:
Euclidian straight-edge/compass constructions,
linear algebra & coordinates, projective geometry,
transformational groups & non-Euclidean geometry.
The most important aspect is that he views geometry from
several different ...