I stumbled upon some tutorial videos on various topic in math by Lorenzo Stadun. I'm sharing the link because a cursory glance indicates that the content is pretty good. So the first thing I ever do with something I like is to criticize it.
In his first pre-calculus review video on logarithms he says $x=\log_a\left(y\right)$ is the inverse exponential of $y=a^x,$ where $y=a^x$ means $y$ is the exponential of $x$.
If someone tells me to write $y$ as the exponential of $x$, I will write $y=\exp\left(x\right)$ which means $y=\exp\left(x\right)=e^x.$ In other words, the exponential, and the exponential function explicitly mean Euler's number $e$ raised to a specified power.
I would say $a^x$ is an exponential expression, meaning that it involves an exponent. I say $y=a^x$ means $y$ equals $a$ raised to the power $x,$ or $a$ (to the) exponent $x$.
What is the standard usage for these terms?
Edit to add:
One place where the exponential function specifically means $y=e^x\iff \ln y=x$ is §8-8 The Exponential Function of Calculus and Analytic Geometry: With Supplementary Problems, Second Edition, by Thomas, George B., 1953. That is the same Thomas as in Thomas's Calculus, 14th Edition.
In that context the exponential function is defined as
$$y=e^x\iff \ln y=x\backepsilon{x\in\mathbb{R}}.$$
Prior to that point $e^q$ was only defined for rational numbers $q$, and was not called the exponential function. Thomas does give the full name as "the exponential function, with $e$ as base and exponent $x$." But continues to discuss $e^x$ as "the exponential function," without qualification.
As was pointed out in the comments, we can always change the base, so all real number expressions involving exponents can be written in the form $e^x$ using the definition $a^u=e^{u\ln{a}},$ which Thomas gives in §8-9 The Function $a^u.$
There is no standard usage for the term exponential by itself, it is usually used with other terms, like exponential growth, exponential function, etc.
The standard usage for exponential function is, as you correctly noted, $e^x$, if nothing is meant by context. However, sometimes $a^x$ may also be called an exponential function for some $a\ne e$ (the point to be noted is that $a^x=e^{x\ln a}$, so one is just a $x$-scaled version of other function).
Exponent, as you again correctly noted, is just the power of a number (for example, $2$ is the exponent and 3 is the base in $3^2$).
Hope it helps:)