Executive summary: Open Cycle Map cannot be regarded as a reliable tool to resolve such marginal cases.
The underlying OSM dataset, which is itself essentially crowdsourced largely using consumer-grade GPS, does not contain elevation data. A few isolated points may have “ele” tags of dubious accuracy, but there is no systematic surveying. It follows that no pure OSM rendering can have contours.
The contours that we see in the Open Cycle Map rendering are a separate layer interpolated from NASA SRTM data. I believe this was done in 2008 and may or may not have been updated since; but the key point is that even if other map features appear to be up to date the contour layer won’t be.
Then there is the thorny issue of vertical datums and geoid models. No method such as GPS or radar which measures the shape of the ground, however accurately, can ever tell you height. Height is determined by gravitational potential, and the shape of the ground is only one factor out of many that determine this. The radar data from SRTM is adjusted to be relative to a global geoid model so that it gives a generally good measure of height world wide. However this is not the same vertical datum as we traditionally use in Britain. Ordnance Datum Newlyn was originally determined by a tide gauge in the last century, but has now been redefined relative to the GPS ellipsoid by applying a local geoid model - this model is more detailed than the global model used for SRTM data.This means that we can expect a variable discrepancy between SRTM heights and ODN heights.
Mean sea level in Britain lies below the global average sea level. I have not done the detailed calculations myself, but I have read that the difference is of the order of 5m, i.e. in general we can expect ODN heights to be around 5m greater than the SRTM data would suggest, though the exact value will vary from place to place. Curiously, this is sufficient to explain the absence of a 360m contour on the map you are looking at, though there are so many other inaccuracies in the process that this may be mere coincidence.
I think the conclusion I draw is that Open Cycle Map is not a sufficiently reliable tool to determine whether the activation area and the access area intersect in this very marginal case.