If we build a setup at the lab, we describe it as home-built when we write the paper. What is the correct word to describe a program written by the author? The program has not been published and there is no intention to do so.
"in-house" program is the term I always use myself and see in other people's papers.
In the corporate context, Wikipedia defines it as
In-house software is a software that is produced by a corporate entity for purpose of using it within the organization.
Here's an example in an academic context
Energy minimization is done with yammp, an in-house molecular mechanics package.
All analyses were done with a battery of in-house codes and external analytical tools.
I wouldn't say "in-house" strictly means "the source code is private". I have sometimes seen people use "in-house" to mean "code our own research group wrote and uses, but which we also make available as open-source code". But if I see a sentence in an article like "All analysis was performed using in-house code", and no reference, then I read it as "We used some code we wrote ourselves which we haven't published anywhere", which sounds like what you want.