I don't think there are ‘official’, legally-defined terms for these relationships; there are far too many subtle differences between one situation and another. But a real-life example of use within one company may help.
I have a client—let’s call it X-USA—which is a regional subsidiary of an international holding company—let’s call that one X-Global. X-Global also owns subsidiaries in other regions: X-SouthAmerica, X-Europe, X-AsiaPacific and X-Africa.
X-Global trades in commodities produced around the world, so these subsidiaries often deal with each other: for instance, X-USA sells US commodities to X-SouthAmerica and buys South American commodities from X-SouthAmerica. In these transactions, the subsidiaries cooperate in some respects: for instance, payments from one subsidiary to another are handled basically by bookkeeping entries at the X-Global level, and of course they are all, in the end, dependent on the overall profitability of X-Global. In some respects, however, they must for legal reasons act ‘at arm’s length’, as if they were unrelated; and X-Global does not direct either subsidiary’s operations in these respects.
Internally, X-USA refers to the other subsidiaries as affiliates.
X-USA also has joint-venture agreements with companies outside the X-Global family; for example, X-USA and another company jointly operate a loading facility for seagoing vessels. These relationships are governed not by the companies’ common ‘parentage’ but by specific contracts between them. Internally, X-USA refers to the other companies as partners.