iPhone 7 never shows "LTE" even on LTE network
Most carriers support LTE on just about any modern smartphone, including iPhones going back to the iPhone 5. Indeed, Three rolled out LTE back in 2014.

Your friend is correct that 4G (HSPA+) is not the same as LTE. In general, LTE has shorter range but enables significantly faster data speeds than plain old 4G (hence its name, Long Term Evolution).

To enable LTE, go to Settings -> Cellular Data Options -> Enable LTE and choose either Data Only or Voice and Data. 



More information:


Manually Enabling LTE on iPhone (cult of mac)
Difference Between 4G and LTE (nyt gadgetwise)
Picture Evidence of 3 LTE (three support)




Edit: Based on OP's evidence, it seems that on 3 UK, LTE is referred to as 4G, so indeed the status bar will never show "LTE", just "4G", even while on a 4G-LTE network. As long as the speeds hold up to LTE-quality, you're probably getting LTE. This is likely a regional difference, determined by the language on your phone.