What's the difference between irradiance and fluence/radiant exposure? A paper$^1$ I'm reading says fluence measures the incoming radiance from all directions and that fluence is similar to irradiance. It's defined by $\phi(x) = \int_{4 \pi} L(x, \vec{\omega'}) d\omega' $ (The tick mark seems to emphasize $\vec{\omega'}$ varies over the integration, rather than being a fixed direction.)

Intuitively, radiance is the amount of light given off in a direction but attenuated by how far off the surface is from being perpendicular to the light direction. $L=\frac{d^2\Phi}{d\omega dA \cos \theta}$

Irradiance is the area light density of a surface. $E=\frac{d\Phi}{dA}=\int_{2 \pi } L_i(\vec{\omega}) \cos \theta d\omega$

Intensity is the amount of light hitting a surface from a particular direction. $I(\vec{\omega})=\frac{d\Phi}{d\omega}$

It also looks like you can write the radiance as irradiance times intensity. 
$L=E\cdot I(\vec{\omega})$

Looking at the formulas, it seems like irradiance sums radiance over the hemisphere over a surface point and attenuated by how much the surface orients toward the light source, while fluence sums up radiance over the whole sphere surrounding a point (to measure inscattering).

Wikipedia says, "radiant fluence is the radiant energy received by a surface per unit area, or equivalently the irradiance of a surface integrated over time of irradiation." But this definition doesn't seem to match the one given above since there's no time involved.

https://www.researchgate.net/post/What_is_the_definition_and_difference_between_light_dose_and_light_fluence_rate

http://omlc.org/education/ece532/class1/irradiance.html


Real-Time Rendering of Translucent Materials with Directional Subsurface Scattering, Alessandro Dal Corso, 2014
I can only cite what I have learned in my lecture on Global Illumination techniques which was unfortunately some time ago:


Radiant Power : The amount of energy emitted by a light source in unit time. denoted by $\Phi$ and is measured in Watt which equals to Joules per second. (This does not specify any area!)
Irradiance: The irradiance denotes the incident radiant energy on a surface. It is defined as $E=\frac{d\Phi}{dA}$ - This quantity denotes the incoming energy per area and is measure ind $\frac{W}{m^2}$ Note here: radiant energy is measured in Watt which is Joules per second!
Radiant exitance: This is the reverse direction of Irradiance and is the energy an area emits in unit time and has the same definiton and units as Irradiance only with reverse direction $M = B = \frac{d\Phi}{dA}$. This is also often called radiosity B.
Radiance: Is the power per projected area per solid angle. It is defined as $L=\frac{d²\Phi}{d \omega * dA * cos\theta}=[\frac{W}{sr*m^2}]$ where $\omega$ denotes the solid angle, $\theta$ denotes the angle between emitting surface normal and the direction the energy is traveling to. So this is not attenuated by the distance but by the angles.


In general as I learned the distance between emitting and receiving surface is only important for Radiance because the projected area will be smaller if they are farther apart.

The time you are missing in your definitions seem to come from the conversion of Watt to Joules Per Second.