Tributary Trade was strictly state-sanctioned and could not be considered "private."
For instance, the foreign tributary delegation should file a report beforehand to the Ming government, listing their numbers and goods, then wait for the Ming side to approve and set up a price for the goods; if the Ming refused to trade, or decided to pay much less for the goods, the envoys cannot reject the deal normally. The first Mongol invasion I mentioned above was triggered by the Ming decision to strike down the horse payment to 1/5 of the original.
Most importantly, the Tributary Trades were more about the Suzerain-Tributary relationship than the trade. During such trades, Ming usually bought the tributary goods much higher than the market price - to show that they were "benevolent" suzerains - and sometimes, they just straightly handed out gold and silver to the delegations. For Ming officials, most of the Tributary Trades were too costly and only had marginal efforts for diplomacy. This is also why the Treasure Fleets were canceled after the Yongle; it was such a money sink that even Ming (then perhaps the richest country in the world) could not afford. Overall, we cannot really say these were market-oriented trades, which is why they require the modifier "tributary."