The key point that I think you may have missed is that the emission line of the transition used to generate photons in any laser has a finite width. In the case of a helium neon laser, 632.8nm refers to the wavelength where the emission is brightest, but it doesn't mean that all the light comes at exactly that wavelength. There's a spectrum with a peak at that wavelength, but the peak has some width meaning that some light is emitted in a band of wavelengths to either side of 632.8nm.
On top of that spectrum, you have the optical cavity modes which act like a filter. The spacing of the longitudinal modes depends on the length of the cavity, and the spectrum of those modes looks like a comb. The peaks of that comb spectrum will generally be narrower than the laser emission peak, and the result is that where the peaks of the comb overlap with the laser emission peak, light at that wavelength will be allowed through.
Depending on the wavelength of a particular laser and the length of the cavity, it's possible to design lasers with multiple cavity modes falling within the emission peak, resulting in light with several distinct frequencies, or you can have lasers with only a single cavity mode falling within the emission peak. These "single mode" lasers are useful in some circumstance because they emit light in a narrow band of wavelengths, and the exact wavelength can even be tuned by varying the propertied of the cavity slightly so as to shift the cavity mode within the emission peak. This can be done either by carefully designing and manufacturing a cavity with a particular length, or for some types of laser it can be done dynamically by heating the laser material to change the length and/or refractive index.