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In this lecture from MIT, the professor defines the intensity of a wave as the square of the amplitude of the wave. But, at the same time, the professor defines the intensity of light as the number of photons per second. Are these definitions equivalent?

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As the Wikipedia page for irradiance explains in detail, there exists a huge range of radiometric measures of light intensity, depending on whether you care about the angle of emission or the spectral distribution over different frequencies or wavelengths, and so on. From these, the irradiance is the most natural measure, and (once you put in the suitable constants) all three versions of it are equivalent.

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Additionally, when I searched up intensity on Wikipedia, I was unable to find a definition of intensity that matched the definition provided in the lecture. Some of the definitions of intensity that I did find on Wikipedia were "power per unit area," "power per unit solid angle," and "luminous flux per unit solid angle."

The parabolic mirror has the property of rendering parallel, or nearly so, the rays from a light-source placed at its focus. From Project Gutenberg.

For a collimated light beam, the most relevant intensity measure is the irradiance, i.e. how much energy passes through a unit cross-sectional area in unit time (i.e. power per unit area), and for a plane wave with electric-field amplitude $E_0$, the irradiance is given by $$ I = \frac{\epsilon_0c}{2}E_0^2. $$ The (average) number of photons that pass through that area per unit time, $R=dN/(dt\:dA)$, is then obtained from the irradiance via $$ R = \frac{I}{h\nu}, $$ i.e. by dividing the energy flux by the energy of each photon as given by the Planck relation $E=h\nu$ from the light's frequency $\nu$.

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Thus, my question is: what do the terms "intensity," "intensity of light," and "intensity of a wave," mean? What are their definitions?

The *energy of a typical wave (like a real ocean wave) is given by the square of its amplitude. But in quantum mechanics however, the probability density of a wave function is given by the amplitudes square. A quantum wave function is NOT the same as a typical wave as we know it in the macroscopic world, though there are similarities mathematically. A wave function is a probability distribution. Here, intensity means frequency of observing the particle in a particular place (probability of finding the particle in a particular place.)

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I know that light has both properties of a particle and properties of a wave, but the two different definitions of intensity have different units, which seems confusing to me.

The term "intensity" has many meanings, and can be misleading depending on who interprets the word. Intensity here probably is intended to mean energy intensity- as in how luminous/energetic that light is. I feel like when your professor says "number of photons per second", they are referring to the frequency of the light wave. Professor was perhaps referring to flux of an EM wave? Its flux can be thought of as the amount of something that passes through or onto something. Like the other answer says, light is a difficult thing to describe, and there are many different properties to measure.

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