There is no special term for this use of we, because there is nothing grammatically or semantically peculiar about it.
‘We need to mow the lawn’ does not mean ‘You need to mow the lawn’. What it means is something like ‘Our family, considered as a group, has the responsibility for mowing the lawn’. Given that the speaker is a member of that group, there is nothing unusual about using we for it. The meaning of the utterance does not specify how the family will discharge this collective responsibility, that is, to whom it will allocate the execution of the physical actions of mowing. It leaves it open whether that will be this or that member of the family, or perhaps somebody from a commercial lawn-mowing service that the family may choose to engage.
Of course, it is possible that when somebody in a particular family says ‘We need to mow the lawn’, there will be an expectation that one specific member of the family (perhaps the one who has always done it before) will do the physical mowing. That is, however, a matter of the practices established within that family; it is not a matter of the meaning of English words.
Similarly, to use the example that appears in another answer on this page, when an officer says ‘We need to take that hill’, he means something like ‘Our unit (considered collectively) needs to take that hill’. Given that the officer is a member of the unit, there is nothing unusual about his referring to the unit as we. The officer does not by this sentence say anything as to which members of the unit will perform which specific actions; that will be specified by the commands that will follow.