The question is do I need 2 pairs, one for the roads and another for the trails/fields? Is it like bikes and the number of shoes is N+1?
Yes. Yes, you do.
These days, 90% of my running is off-road: trail and hill mostly.
My trail shoes are Salomon Speedcross 5 ( and previously the Speedcross 4 ).
They make a huge difference to your grip in soft ground and mud.
But the thickness of the soles is taken up more by the lugs, so the cushioning is a bit less.
So I try to avoid hard surfaces in them, favouring the grass verges. This also protects them from getting worn down on the tarmac.
They don't have great grip on wet slick rock, or ice however.
They are also a bit heavier than road shoes, so you'll be a wee bit slower in them where the extra grip is not helping.
Same as a mountain bike versus a road bike really!