You are failing to see the difference between the length of day in state and independent..
Boarding schools pay their staff to teach, but also to carry out a wide range of other duties which take up considerable time. So a full time teacher won't just teach and prepare lessons but also spend lots of time as a tutor, , running teams and doing lots of other things, often working 6 days a week (or more) and being available on and off until far beyond the end of lesson times.Because of this, it is very difficult to be precise about exactly how many lessons spent teaching are full time, because it will be adjusted according to other responsibilities people have. Anyone who doesn't undertake house duties or work at the weekend or do extra curricular is doing less than those who do, so will be paid less.
There is no point feeling furious about this. If you don't understand how it is all calculated, email them and politely ask - without being stroppy. Simply ask how they calculate the %pay for part timers and how they calculate the no of teaching periods and non contact time. And they will then explain it to you.
You may well find that they then ask you to do other stuff too such as run matches on a Saturday as well on a pro rata basis - and if you aren't doing those, then you are contributing less and being paid less too.
In independent schools , there is a need for a lot of flexibility and goodwill, rather than rigidity. In term time, people sacrifice a lot and often work very long days, with blocks of non-contact time at odd parts in the day. Some people like this because the pace of the day can be calmer, but it is also longer.
Not having a Form, not running after school activities and not running Saturday fixtures will save you more time than the few free periods you seem keen to have paid.
I would ask. And I would also consider carefully if this set-up is for you. You don't sound at all flexible and also extremely grumpy, without even really seeking to clarify the arrangement. If you want ridgid and working to rule, this won't be for you.