Taking the wheel just makes it harder for someone to ride off - you can still throw it in a van, or carry it.
As you said, there are 2 completely different types of theft: opportunity and planned. The opportunists will take whatever is easiest at the time and will potentially dump it a block later when they find a nicer one. Making it hard to remove or unrideable will put them off. I figure if I need to leave the bike for any period of time, taking the axles with me will cause a would be thief embarassing problems - it'll look like it'll ride but will fall apart quickly, if they get round the lock.
Hiding the make/model won't help much because they'll almost certainly put it on craigs list as "bike $100" whatever is on it.
For the planned thefts, it may make it a bit harder to sell on or hide the more unique it is, but a seasoned bike thief will still be interested in a beater bike with high end components on it. Making it harder to steal will help, and making it harder to find will also help (so taking different routes home, being aware of being followed, using Strava privacy zones, etc).
I always try and make my bikes a bit unique with some stickers and non-standard components, so that if it does go missing it'll be easier to identify. They are all registered with a bike register which also has stickers.
The other thing I've done is buy a cheaper bike to use on rides where I may need to leave it unattended or go anywhere particularly risky. It's still a nice bike (105 groupset) so if I was leaving it in a city centre for any period of time I'd probably get an almost disposable bike too, like a $100 hybrid.
Finally, insurance. If someone is determined to steal your bike, they likely will and I'd rather they got it cleanly than destroyed my garage or car in the process. So insurance on the bikes I can't afford to lose will get me back to where I was, though you need to be really careful you meet all of the criteria which can be pretty severe.