As much as it might jar with some, London Underground is a heavy rail system. Definitely a metro, but not light rail.
The infrastructure, rolling stock and operating practices do not conform to light rail characteristics. Mainline TOCs run passenger trains over LUL infrastructure, Class 66s were used on BTR jobs for a decade up until this year, and Class 20s were used to deliver S Stock to Neasden - not to mention heritage operations using mainline steam, diesel, coaching stock over the years (and not just on the Met).
Rolling stock, while obviously rated for lower maximum speeds, has nothing in common with light rail vehicles. 1973 and D Stock use the same braking system and traction control/auxiliary supply methods as mainline 3rd rail EMUs of a similar age. Conventional stock prior to the C69s was similarly comparable to the slammers on the Southern in terms of braking and traction control. Continuing the trend, S Stock and 378s share a number of components - presumably so do other members of the Bombardier EMU family.
The only thing which detracts from LUL's status as a heavy railway, albeit tenuously, is the loading gauge of the tube lines.