"go / goes to" $(x \to y)$ vs. "maps to" $(x \mapsto y)$ Is there a convention for determining when to use $\to$ vs when to use $\mapsto$? Or is there some flexibility between the two?

I have only ever seen $\to$ used within the notation for limits (ie I have never seen anything like $\lim_{x {\color{red}\mapsto} a^+}\limits f(x)$) - which makes sense to me.
However, I have seen both symbols used for reassignment of a variable -- eg if a question is posed in terms of variables that may shroud the (otherwise familiar) structure of an equation to new initiates:


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*Ex 1: to help a student familiar with spherical coordinates recognize that $\vec{r} = \left< 5\sin u \cos t, 5 \sin u \sin t, 5\cos u \right>$ for $u \in [0,\pi]$, $t \in [0, 2\pi]$ is the equation of a sphere in polar coordinates by re-writing the expression in terms of the more familiar spherical coordinates $\phi$ and $\theta$, I've seen both "let $u \to \phi$ and $t \to \theta$" as well as "let $u \mapsto \phi$ and $t \mapsto \theta$."

*Ex 2: when making a substitution (e.g. during integration) I've seen "let $u = x^2$," as well as "let $u \mapsto x^2$," as well as "$x \mapsto x^2$."

I realize that there are hardly any universal conventions; so am equally interested in the reasoning for any specific one (i.e. if you really think they should never be interchanged, I'd love to know why).

Context / Motivation:
Most of my (post-secondary) math education has been informal -- a haphazard dipping into many different textbooks; course notes from different universities; online course etc. to complement the basic education I received in applied math during my physics undergraduate degree -- and so I've encountered many different uses of notation. It hasn't been a stumbling block for my understanding (at least, not that I'm aware of), but it feels like high time for me to look into these details. Thanks in advance for your time and help. 
 A: Well, in view of mappings, the notation is, e.g., $\exp:{\Bbb R}\rightarrow{\Bbb R}_{>0}: x\mapsto e^x$. The symbol $\mapsto$ is only used for the assignment of elements, while the symbol $\rightarrow$ has a broader use.
A: Let $f$ be a mapping from two sets, $A$ and $B$. Then, we write $f: A\rightarrow B$. In other words, $\rightarrow$ denotes the domain and codomain of a particular mapping. 
On the other hand, we use $\mapsto$ to denote what an arbitrary element of the domain explicitly maps to. In other words, we write $f: x\mapsto f(x)$ to show that an element $x\in A$ maps to $f(x)\in B$. 
As you see in your two example, $\mapsto$ is indeed the correct notation, because we are specificying the mapping under which arbitrary elements are changing. 
Broadly, $\rightarrow$ is used to show which two sets the mapping $f$ is between, and $\mapsto$ is used to show what the actually “rule” of the mapping is. 
As you observe, $\rightarrow$ is also used in limits. This is a separate usage of this symbol,  where roughly $x\rightarrow a$ means “as $x$ tends towards $a$.” This is a distinct and separate usage from the main two usages of $\rightarrow$ and $\mapsto$ that I discuss above. 
