It is not correct that a private pilot can be reimbursed for expenses. Pilots are permitted to share expenses for a common purpose.
The long standing (~30 years) rule is the pilot's share can't be less than their proportionally equal share, based on the total number of people in the aircraft. The pilot can't walk away from a private flight with a net $0 cost or it isn't really a private flight, the pilot is being compensated. And the passengers and the pilot need to be engaged in a common purpose activity, ergo it cannot just be about transporting people from A to B if the pilot has no need himself to go from point A to point B.
If the purpose of the flight is to test an app for the company, it is unquestionably a commercial flight and the pilots are employees for the app company. Depending on myriad other factors, the size of the aircraft and how many passengers are on board, a two pilot crew rule might apply and if so absolutely a student pilot does not qualify.