Who pays the Piper plays .........!
The simplest way to look at this is to consider the purpose of the flight. From what is desribed the purpose is to undertake an aerial survey. To pilot an aircraft for such a purpose on behalf of other people it is 'aerial work' irrespective of whether money changes hands or not. If others are carried also for the purpose it becomes the public transport of passengers.
I mention "on behalf of other people" because should you be a surveyor and flying yourself to do the survey for your own purposes then it would not be aerial work. In this case you would be using the aeroplane in the same way that you might otherwise use your car. Using the aeroplane musn't in itself be a requirement to do the job otherwise the actual flight is for the purpose of 'aerial work' or 'public transport'.
An example of non-public transport/Aerial work would be; you and three others from your office need to be at a certain place by a certain time. You elect to hire an aeroplane with you as pilot, this being a quicker and preferred method to driving. It is clearly a private flight and you take no benefit for you had to go anyway.