Just so we're clear on this:
- My expressed desire to buy 10 shares of AAPL is called a bid
- Your expressed desire to sell 10 shares of AAPL is called an offer (or ask)
The hypernym for bid and offer is quote, and the instruction by which you express this quote is an order; but, critically, n̲o̲n̲e̲ ̲o̲f̲ ̲t̲h̲e̲s̲e̲ ̲t̲h̲i̲n̲g̲s̲ ̲i̲s̲ ̲a̲ ̲t̲r̲a̲d̲e̲.
So what do you call it when my bid (order to buy) matches your offer (order to sell) such that I get your stock, and you get my cash?
That is the trade. The word you're looking for is trade. The trade is distinguished from the earlier actions (bidding, offering) because those are quotes: desires, intents; the trade is the transaction.
Therefore, there is no need for additional adjectives, or a different word, for a "done" trade or a "closed" trade; a trade is, by definition, closed and done.
Aside: you will sometimes hear "closed trade" in the securities industry, but here "trade" is used metaphorically or figuratively to mean "trading idea", or "strategy", and is technically a closed "position", which takes two actual, literal, trades to effect (one to open the position, and the other to close the position, and capture the profit or loss).
Also, in the specific scenario you're describing, where one student in your class is trading with another student in your class, and neither one has to go to the open market to find a counterparty, what you've got is an internal cross or just cross (which is done -- you guessed it -- through internalization).
In the US, all trades, whether internalized (crossed) or done on a public exchange or ATS (where orders are said to be matched as opposed to crossed), are still required to be reported to the public, and so your students' internal trade will still show up on the tape as a tick.