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I would use Cambridge. I have taken a few language proficiency tests. Mostly due to the interview stage of applying for TEFL work, as well as to apply for my TEFL course. I have taken the Oxford test as well. My proficiency level is C2 each time. I tried both of your links, and got C2 on Cambridge, but C1 on the British Council.
My decision isn't based on ego, but based upon analytical reason. If one has a constant score of "High Proficiency", Then a level has been set. So if one has scored under that level in one test, it must be that the test is faulty. I am not saying I am right, but I like to analyze things before making decisions. Plus Oxford and Cambridge kind of set a high level in English proficiency, So I would lean towards their thoughts, rather than the British Council.