Lettering is not a font or how to understand the terminology of typography

Font – a set of characters of a certain size and pattern, intended for reuse many times and in any order.

Lettering is not a font. Lettering is a one-of-a-kind letter made (drawn, cut) for one specific situation. Many people call a font anything if there are several letters. However, often what they see is lettering. Its main difference is the emphasis on the graphic component: unique forms that combine with each other in a particular word or phrase.
The difference can be seen on the motivational posters from the site blck.ink

A printer is not necessarily a font designer

This is a person who understands the art of printing, a specialist in typography and book design. Typographers include fonts, graphic designers and other professionals.

Font designer creates fonts. Such a specialist is very familiar with the history of the font, as well as the history of styles, the history of graphic design and advertising. He is sure to draw by hand, knows the composition and psychology of visual perception. Has graphic computer editors and specialized programs for creating fonts.
A headset is not a font

Back in 1898, Morris Fuller Benton, working as a lead artist in the American company “American Typefounders”, introduced the concept of “headset”. More than a century has passed, and the vast majority of people do not know what the difference is.

A typeface is a set of fonts, united by stylistic unity and consisting of a certain set of typographic characters. Each headset must have its own name.

The headset usually contains alphanumeric and punctuation marks and special characters. There are also headsets that consist entirely of non-alphabetic symbols, such as those containing mathematical or cartographic symbols.

A font is a drawing in graphics, a set of letters and signs, which together represent a coherent system, both in stylized and compositional terms.