A font is a typeface in a specific size. A glyph is a vector shape, and in your context it is [the vector shape of] a specific character from a typeface in no particular size.
A comment below suggests that digital fonts can have multiple sizes.
The word "font" is from French fonte, from Middle French, act of founding, from Vulgar Latin *fundita, feminine of funditus, past participle of Latin fundere to found or pour (in the sense of casting). It refers specifically to the founding of a typeface in lead. As pointed out by the commenter, this is necessarily in a particular size.
In digital typography, font refers to a rendering of a typeface at a particular pixel size. This non-trivial process involves hinting, alpha blending and pixel snapping according to complex rules and quite a lot of metadata.
In days of yore this was a bit much to ask for in real time and typefaces were pre-rendered at various sizes into font files. Now that rendering is done in real time on the video card, you don't install fonts, you install typefaces, but for "consistency" Apple and Microsoft (and the Linux people, for that matter) refer to them as fonts in typography management user interfaces, leading to a widespread misperception of the meaning of the word font. The font picker dialog on any given platform actually does pick a font, because it specifies a particular size.
To all you who say "language belongs to the people" I say that if an entire nation of people call a dog a cow, it still won't give milk.