Correct - usually the hood is a separate piece tucked into the neck of your suit. Your suit will have a zipper somewhere, if its a 1-piece full suit, generally a the back that can let in a trickle of cold water.
The attached hood prevents the trickle of cold air. You are correct that the hood cannot be removed, and I guess it is a bother when out of the water. Note - a fullsuit w attached hood is a also bit harder to don/doff - the zipper is across the chest. If im out of the water and getting warm, I tend to either slide my head out of the face hole (a bit uncomfortable) or unzip and remove the hood so that my upper torso is free of the suit (also a bit uncomfortable)
I suppose that a farmer john has the zipper in the front, not the back. I don't recall the "trickle of cold" from a farmer john, so maybe that isn't an issue. However, I only wore a farmer john a few times long ago before I got my own suit.
My other issue with farmer johns is they generally mean more lead.
One note - I don't know how effective only wearing the farmer john jacket would be, as its designed to fit over the overalls and may be too loose to be used on its own.
That's one reason I suggest also suggest a fullsuit with a vest with attached hood over it -- for maximum cold protection. And if you are somewhere warmer, you can change components
Lots of suggestions from different folks. Ultimately you will choose what suits you best. It may be that what works for me in NorCal (down to 48 deg F, altho not lately) may not be warm enough for you in the Great North, so a farmer john is a better option.
Also, one other thing I haven't thought of, I cant tell from your post if you are male or female, and I could see a female having a different experience re fullsuits w/ attached hoods.