It depends on what you mean by "program".
First, the excerpt from the video is misleading when it says "That session starts with one job, containing the shell process." As explained in What is the difference between a job and a process? and also as implied by the POSIX definition of job, a job
is a shell concept. So a running shell is not itself a job unless it is part of a job from a parent shell process. In the excerpt, the parent process is the terminal which, as explained in the The Bash Guide, is just a graphical interface program used to run text-based interface programs which aren't necessarily shells.
Here are some commands to illustrate that the table from The TTY demystified is accurate:
+ Stopped cat
$ cat | sort
- Stopped cat
+ Stopped cat | sort
You can see in the last output that two jobs exist:
cat | sort, the later being equivalent to the
ls | sort from the table.
As to whether "a program run by bash runs in a new job", POSIX defines:
A set of processes, comprising a shell pipeline, and any processes descended from it, that are all in the same process group.
So if you consider that "a set of processes, comprising a shell pipeline" is a running program (which you can on the basis that a program is a set of instruction to be executed), then yes.
But since "program" is a more general concept than just "shell pipelines", then it is not generally true that programs run by bash run in a new job. For example
sort are also programs but they don't each run in a separate job.
Also, shell builtins are "programs" in the sense that they are "set of instructions". But they are embedded components of the shell program itself and so they do not necessarily need the creation of separate child processes to be executed. In circumstances where they are not executed in separate child processes they cannot be part of a job for the shell.