You've not given us enough information to answer the question, as you've not said which Canary Wharf station you're entering, nor have you said what route you'd be taking.
However the
single fare finder makes it clear that the £1.50 fare is for "Avoiding Zone 1" and requires use of a pink route validator.
(Yes, there are occasions when you may be charge a fare that assumes you avoided Zone 1 when you did go via Zone 1, but let's not get into that now)
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People generally (depending on context) refer to the big LU trains as "the tube" even though they're not tube trains and don't run in tubes. This is because LU is known colloquially as "the tube" even though the original lines built by the proper railway companies that pre-date LU were cut-and-cover.
(But let's not go any further than that on this thread
)
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Fares are sometimes charged based on assumptions; sometimes the assumption has to be that you will use the cheaper of the routes.
Also
some NR routes are charged at TfL rates (LO is charged at the
cheaper of the two rates), so it's not necessarily a case of a particular TfL rate fare being for travel "exclusively" by TfL sponsored services. For example, the use of Greater Anglia between Stratford & Liverpool St would
not invoke NR rate charging.