What's the difference between irradiance and fluence/radiant exposure?
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.