As a sibling comment mentions, these commands were (are) commonly composed into scripts. As the name implies, however, a script is just a playbook for a series of commands to run. Given the terminals of the era, I'm sure short commands/variables/etc. were appreciated in scripts as well, but it seems to me that the primary motivation for optimizing input speed would be the use of these commands in an interactive environment.
A few examples of these core short program names: ls, cat, cp, rm, wc, uniq, cmp, diff, od, dd, tail, tr, etc.
Shell scripts have been part of UNIX since day 1. A lot of early UNIX commands were implemented as scripts, and shell was always intended to be one of the main extension mechanisms for users. Scripting is fundamentally based on the assumption that a script will run various commands, glued together in a user-specified way.
So it's true to say that most early UNIX commands were designed to be useful in a script, which implies that they were designed for use by programs. Maybe not designed _exclusively_ for use by programs, but definitely an important design consideration.
> Traditionally, UNIX commands were written under the assumption they were going to be used primarily by other programs
Is there any real factual basis for these claims? I find them hard to believe, considering the origins of UNIX and the fact that shell was the primary (or even only) user interface for the system, and it was pretty much from day 1 designed as an interactive system (contrasted to the more batch/system oriented mainframes of IBM etc)