In this material, it gives the definition of modes:
The modes are basically defined by the properties of coherence and orthogonality: modes are orthogonal solutions of the wave equation.
Since the modes are eigenfunctions in this material, why do we need to use a new term 'mode' instead of the original one, i.e., eigenfunctions?
Modes are governed by eigenfunctions, I agree. In quantum optics, we need more than just eigenfunctions to describe a state of light: we need to know how many photons have properties corresponding to each eigenfunction. This is somewhat beyond what an eigenfunction describes, so we need a new term.
For example, we can have a two-photon state of light where each photon is in a different spatial mode. That means that the light does not correspond to a single eigenfunction, nor does it correspond to a superposition of two eigenfunctions. We need a new way to describe it; hence, modes.
We could also have a two-photon state of light where both photons are in the same spatial mode. How could this be expressed if all we have are eigenfunctions? Specifically, how would we distinguish between this state and a single-photon state of light corresponding to the same eigenfunction? If you say we could describe states of light by their eigenfunction and by the average energy, that would not be sufficient, because there are multiple states with a single eigenfunction that all have the same average energy.
In all, we could simply say that light is described by a bunch of orthogonal eigenfunctions and extend the definition of eigenfunctions to include a description of the Fock space associated with these eigenfunctions. Since eigenfunctions are very well defined in the rest of mathematics, it is eventually easier to define a new term, modes, that can incorporate both the properties of eigenfunctions and the properties of the Fock space describing the photon-number superpositions for all of the eigenfunctions.
In math: label two eigenfunctions by $a$ and $b$, a single photon with a single eigenfunction might be written as $|1\rangle_a\otimes |0\rangle_b$ or $|0\rangle_a\otimes |1\rangle_b$. The eigenfunction language can't really distinguish between, say, $|1\rangle_a\otimes |0\rangle_b$ and $|2\rangle_a\otimes |0\rangle_b$. Moreover, it does not do well at describing $|1\rangle_a\otimes |1\rangle_b$. So we either add to the definition of eigenfunctions, or we define a new term that lets us write $$|\psi\rangle=\sum_{i,j}\psi_{i,j}|i\rangle_a\otimes |j\rangle_b.$$
