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Your interpretation is incorrect.
Each card has a triggered ability. The current oracle rules are:
So both triggered abilities go on the stack when the non-token Wurmcoil Engine dies. What is confusing you is that they do not immediately resolve. (See section 603 of the Comprehensive Rules.) Technically, whoever has priority puts their triggered ability on the stack first, but in this case, it doesn't matter much.
Therefore, the controller of the Wurmcoil Engine gets the two artifact tokens and the controller of the Mimic Vat may have the Wurmcoil Engine imprinted on their Mimic Vat. If they choose to do so, the Wurmcoil Engine card is exiled and any other imprinted card is returned to the owner's graveyard.
If and when a Wurmcoil Engine dies and goes to the graveyard, its ability will trigger and go on the stack. If it was a non-token creature at the time it died, Mimics Vat will also trigger at the same time and go on the stack. The active player decides the order in which they are put on the stack, but either way, both effects will resolve unless countered. Meaning: for a non-token Wurmcoil Engine, 2 tokens will be created and it can be exiled with the Vat.
A Wurmcoil Engine (or a token copy thereof) that gets exiled neither gives tokens nor triggers the Vat. However, if you have a way to destroy or sacc a Wurmcoil Engine token before it is exiled, the token Engine will also create its 2 smaller tokens, though still not trigger the Vat.
The players gets the tokens. The key here is that both the abilities are triggered abilities and no replacement effects. If it would have been worded something like: "If a creature would be put into your graveyard from play you may exile it instead. If you do imprint that creature on ~this~.". Then the creature would never be put into the graveyard and the creature's "When put into graveyard" would not trigger.