注文 and 購入: placing an order and buying an item
Track ordering, purchasing, and delivery separately, without assuming that a purchased item has already been received.
Requesting goods or services
注文, read ちゅうもん, means ordering goods or services. 店に新しい部品を注文しました。 means that new parts were ordered from the shop.
注文しましたが、在庫を確認しているところです。 says that an order was placed but stock is currently being checked. It does not establish completed shipment or delivery.
Stating that an item was bought
購入, read こうにゅう, means buying an item. 仕事で使う机を購入しました。 says that a desk for work was purchased.
購入した机は来週届く予定です。 says that the purchased desk is expected to arrive next week. Purchase and receipt need not occur at the same time.
Reading the sequence of transaction and receipt
注文の内容を変更したいです。 requests a change to an order. Expressing the desired change does not mean that it has been processed.
購入前にサイズを確認してください。 asks someone to check the size before purchasing. Missing 前に would turn this into a different sequence.
Has the desk already been received in 購入した机は来週届く予定です。?
Show the answer and why
Receipt is not stated. The desk was purchased, and its arrival is scheduled for next week.