Why Does Finitely Generated Mean A Different Thing For Algebras? I've always wondered why finitely generated modules are of form 
$$M=Ra_1+\dots+Ra_n$$
while finitely generated algebras have form 
$$R=k[a_1,\dots, a_n]$$
and finite algebras have form 
$$R=ka_1+\dots +ka_n$$
It seems to me that this is an flagrant abuse of nomenclature, and has certainly confused me in the past. Is there some historical reason for this terminology? Any references to the genesis of these terms would be greatly appreciated!

The terminology is actually very appropriate and precise. Consider that "A is a finitely generated X" means "there exists a finite set G such that A is the smallest X containing G".
Looking at your examples, suppose $M$ is a finitely generated module, generated by $a_1,\dots,a_n$. Then $M$ contains $a_1,\dots,a_n$. Since it is a module, it must contain all elements of the form $Ra_i$ and their sums, so it must contain the module $Ra_1+\dots+Ra_n$. However, since this latter object is in fact a module, $M$ need not contain anything else and is in fact equal to this module.
If $R$ is a finitely generated algebra, we can go through the same procedure as before. However, since algebras have an additional operation (multiplication), we must allow not only sums of elements of the form $ka_n$ but also their products. This gives us that $R$ must contain all polynomial expressions in the elements $a_1,\dots,a_n$, i.e. it must contain the algebra $k[a_1,\dots,a_n]$. Again, since this latter object is in fact an algebra, $R$ need not contain anything else and is equal to this algebra.
A finite algebra seems to be a name for an algebra which is finitely generated as a module. Your example is then consistent with what I wrote above. I do admit that the name seems somewhat misleading.
