I suspect engine here is simply synecdoche or ellipsis for the entire engine bay or engine compartment— the entire space "under the [AmE hood / BrE bonnet]" where the engine block and its associated components, electrical supply, cooling system, fluid tanks, and so on sit.
Such usage is evident, for example, throughout this 2007 New York Times article about a different unwelcome passenger:
As if New York City car owners don’t already endure enough indignities… it turns out that rats, of which the city has an ample supply, love to cozy up inside car engines this time of year.
While one might imagine a snake finding its way into an intake or a disconnected hose, no one would ever make such a mistake about a rat— least of all a New York City rat— in a nest. The author is clearly talking about the spaces around the engine itself.
The Times is aimed at a general audience, which (particularly in New York City) may not be particularly careful about automotive terminology. Someone who works in the field uses more specific language—
“They like to go into the engine’s compartment to stay warm and they build a nest there,” said Gus Kerkoulas, the owner of Z P Auto on Great Jones Street in Greenwich Village.…
— whereas the author of the article uses engine and engine compartment interchangeably:
One solution, Mr. Kerkoulas said, is two socks filled with moth balls, an old farmer’s trick. Hang them in the engine — away from any moving parts — and that will deter the rats, he declared.
Additionally, I would note that hide is used with prepositions which indicate location, not motion. Thus, your guest cannot hide into, hide towards, or hide back to even the most spacious engine bay, but can slither, scuttle, swarm, etc. into it to hide in or inside or within it.
The closest fit is 2.; brain cells (neurons) are connected to each other and can send electrical signals via nerve fibres (axons). These signals carry information, just like in a wired computer network.
'Wired differently' is figurative speech (of course, everyone's brain cells are different from the next person), and it indeed means your way of thinking is different from what is normal. (Depending on the context, this can even be a derogatory term.)
Best Answer
It's an innovation. There are no results between 1800-2000 in Google Books Ngram Viewer, and there are likewise no results between 1810 and 2009 in the Corpus of Historical American English. The OED does have a brief entry for entask which says simply "see en- prefix", but since the word essentially does not exist in modern English, it's likely that this new use has been independently re-derived in the same fashion.
It clearly means "to be given a task":
Personally, I don't think this qualifies as a standard derivation, and I'd suggest that learners of English avoid this word, as others may share my judgment. It sounds hokey.