I know that there are various bands referenced in meters (or cm). My question refers to what exactly the word "meter" is referring to when identifying the 5MHz band? I know that the 60-meter band is 5MHz, and considering that communications on that band can go farther than 60 meters, I am assuming that "meter" refers to the wavelength. Is that assumption correct?
That's right, the 5 MHz band has approximately a 60 m wavelength. Back in the early days of radio people often measured where they were on the airwaves by wavelength rather than frequency. Nowadays the radio display shows the frequency, but amateurs still refer to bands by their approximate wavelength.
Wavelength x frequency = speed of light, 2.99792e8 m/s. Or to put it another way, wavelength = 299.792 / frequency in MHz.
The wavelength also literally defines the length of a full-wave resonant wire antenna -- and specifies where along the wire the voltage will have peaks and valleys! So it has practical value when constructing an antenna, which is probably why amateurs still use it.