Authorship for papers not completed before leaving, where I am major contributor?
If you only did programming support for a project, and did not design or collect data, or design the experiment, and did not have a role in writing the manuscript, you probably qualify for only an acknowledgement, and not full authorship. I'm not clear what you mean by "feasibility research" - it's possible that work would qualify if it went beyond normal software design processes (i.e., are you determining whether the idea was feasible for YOU to accomplish in a certain time frame, or feasible at all, and what was the extent of other approaches that you considered...given the time frame you state of 15-20 days work, I am very skeptical there was extensive feasibility research necessary).

15-20 days work is a miniscule amount of time for a typical scientific publication - I would say the normal range is 6 months to 2 years. This range counts all of the failures and false starts inherent in science, refining ideas, etc. Though it is possible all of the data and analysis that goes into a paper takes place on a shorter time frame, it is unfair to exclude all of the productive failure along the way.

Programming by itself isn't really a direct contribution to a scientific project, regardless of the time spent, unless you are developing algorithms, or doing something novel (for example, optimizing a program to the extent that a previously computationally intractable question can be solved in a reasonable timeframe).

I doubt your PI actually does not know the conventions of authorship in the field; it seems more likely that they knew you were overestimating your role in the project, gave you the appropriate credit, and didn't want to argue with you over it further.

More generally, and not speaking to your situation specifically, leaving a lab should not affect your claim to authorship on any work you did while you were a member. However, meeting all the requirements for authorship will indeed require further, unpaid work. At a minimum, this would include review of the manuscript before submission and publication, and in most cases would include substantial writing/editing support, preparation of final figures, etc.