The question doesn't make much sense because when you ask for associative commutative binary operations on the set of real numbers, you forget the whole structure of $\mathbb{R}$. It is just a set, and for the question only the cardinality matters, which is known as the continuum $c$. If $X$ is any set with cardinality $c$ and some associative commutative binary operation, we get a corresponding one on the set of real numbers. For example the set of integer sequences has cardinality $c$, and we can take its addition, or its multiplication.
I would not encourage you to introduce a new terminology, for two reasons.
First, it would increase the confusion between existing terminologies (see below). Secondly, it could make it difficult to find relevant information.
There is a large litterature on Semigroups. The free semigroup on a set $A$ is denoted by $A^+$.
Idempotent semigroups have been studied for a long time and bands is another well established terminology for them. In particular, it is known that every finitely generated free idempotent semigroup is finite (a nontrivial fact, as emphasized by Andreas Blass' example, see [3] for an efficient algorithm). Moreover, a complete classification of the varieties of idempotent semigroups is available [1].
Commutative semigroups are also well studied, [2] is an excellent reference. Idempotent and commutative semigroups are also known as semilattices. The free commutative semigroup on a set $X$ is denoted by $F_X$ in [2], but this is a context-depending notation: $F_X$ or $F(X)$ could be used for the free object on $X$ for any algebra.
Magmas are sometimes called groupoids. See your own question for a notation of the corresponding free algebra. Idempotent magma is a very natural name: it is used for instance in two answers to this question. Commutative magmas have their own wikipedia entry (rock, paper, scissors being the emblematic example). Commutative and idempotent magmas are used in this thesis.
[1] J. A. Gerhard, (1970), The lattice of equational classes of idempotent semigroups", Journal of Algebra, 15 (2): 195–224
[2] P. A. Grillet, (2001), Commutative Semigroups, Springer Verlag, ISBN 978-0-7923-7067-3
[3] J. Radoszewski, W. Rytter, Efficient Testing of Equivalence of Words in a Free Idempotent Semigroup. SOFSEM 2010: Theory and Practice of Computer Science. SOFSEM LNCS 5901, Springer (2010) 663-671.
Best Answer
"I doubt the list is big."
That, I'm afraid, is quite wrong.
What you are straining for is Universal Algebra, which seeks to study the general features of algebras.
Given a set $X$, and a nonnegative integer $n$, an $n$-ary operation on $X$ is a function $X^n\to X$. You can also have infinitary operations, where you take an infinite ordinal $\alpha$ and consider a function $X^{\alpha}\to X$. Given an operation $f$ on $X$, its arity is the $n$ such that $f$ is a function from $X^n$ to $X$. So the arity of addition in $\mathbb{R}$ is $2$, for example, since it is a function $\mathbb{R}^2\to\mathbb{R}$, mapping $(a,b)$ to $a+b$.
A type is an ordered pair $(\Omega, \mathrm{ari}_{\Omega})$, where $\Omega$ is a set, and $\mathrm{ari}_{\Omega}$ is a map from $\Omega$ to the ordinals. The elements of $\Omega$ are the operation symbols, and the ordinal $\mathrm{ari}_{\omega}(s)$ is the arity of $s$ for each $s\in \Omega$.
If $(\Omega,\mathrm{ari}_{\Omega})$ is a type, then an algebra of type $\Omega$ is a pair $(A, (s_A)_{s\in\Omega})$, where $A$ is a set, and for each $s\in \Omega$, $s_A$ is an $\mathrm{ari}_{\Omega}(s)$-ary operation on $A$.
For instance, a type $(\{+\},\mathrm{ari})$, with $\mathrm{ari}(+)=2$ defines algebras that consist of a set with a binary operation.
Once you have a type, you can impose identities. For instance, in an algebra as above, you can impose the condition that $+(+(a,b),c) = +(a,+(b,c))$, which is associativity, etc.
The collection of all algebras of given type $(\Omega,\mathrm{ari}_{\Omega})$ form a category. The collection of all algebras of a given type that satisfy a given set of identities is called a variety of ($\Omega$-)algebras. The study of varieties of algebras is one of the main concerns of Universal Algebra.
There are too many of them to discuss individually. In fact, if you just consider how many classes of groups there are that satisfy certain sets of identities, the answer is uncountably many (there are uncountably many varieties of groups); in fact, there are uncountably many varieties of solvable groups (though only countably many varieties of abelian groups).
I recommend George Bergman's An Invitation to General Algebra and Universal Constructions, Springer Verlag Universitext; or Ralph McKenzie, George McNulty, and Walter Taylor's Algebras, Lattices, Varieties volume 1, Wadsworth and Brooks/Cole; or George Gratzer's Universal Algebra, Springer Verlag.