But, what is exactly a bitcoin node ?
Is it a client like Bitcoin-qt or is it only a miner ? Or both ?
Bitcoin node refers to a "full" client. A "full" client is a client that owns the block chain and that is sharing blocks and transaction across the network. In opposite a Lightweight client can not be considered as a node because he doesn't share the block chain with the network.
Bitcoin network uses a client to client network infrastructure so there is no difference between a mining client and a non-mining client, they don't have any privileges.