Yes! 'Te rugăm' and 'Vă rugăm' are respectively the informal and formal ways to say 'please' if you are speaking on behalf of a group. If anyone speaks Spanish you'll recognise it from 'les rogamos'...
So... If I understood the above explanations correctly, with 'te' it is informal, with 'vă' it is formal/polite.
And 'rog' is individual to individual, whereas 'rugăm' is individual on behalf of a group. Yes?
Exactly. Actually „vă rugăm” can be either the polite form addressed to a single individual the form to address a group. „Te rugăm” is informal and addressed clearly to an individual (on behalf of a group, all these forms).