My reasons for saying there is no relationship between the .270 and the .280 family are:
Purpose: the .270 directly emerged from the Small Arms Calibre Panel report, which recommended a .27 calibre (or .25 if tungsten was used) cartridge with a flat trajectory out to 600 yards and a maximum effective range of about 800 yards. The .280 was developed as a long-range cartridge able to replace the .303/Vickers combination as well being used in rifles (the US Army was only interested in a cartridge which could match the .30 US round out to 2,000 yards, which the .270 couldn’t deliver).
Ballistics: in accordance with the flat-trajectory medium-range requirement, the .270 ended up with a relatively light bullet of 100 grains fired at a respectable 2,750 fps. In contrast, to achieve the long range requirement the .280 settled on a 140 grain bullet, initially fired at only 2,415 fps.
Case dimensions: The disappointing trajectory of the .280 meant that various longer cases were developed to push up the MV, but these were relatively minor changes in terms of case design (there was also a small adjustment to the rim shape made early on, to match the US .30). The smaller case diameter of the .270 is a much more fundamental difference.
So to sum up, the .270 was the outcome of careful deliberation as a result of WW2 battle experience, while the .280 seems to have been conceived quite separately, to meet the US requirement for longer range. The .270 and .280 cartridge projects were actually run in parallel for a brief period, before the .280 was chosen.
Nice photo, by the way!