Usually, unless the plan calls for a "net" size, then it nominal.
So a 2" x 6" nominal will normally only be about 1 1/2" x 5 1/2" after it has been planed and jointed - "dressed on 4 sides" (D4S)
HOWEVER, when talking about a 1" x whatever, the thickness can be a little less, but the width should be exactly what is called for in the plans - so a 1" x 8" could be 3/4" by 8" which would be ok, but 3/4" x 7 1/2" would probably not be ok.
The designer will normally have taken that into account when specifying the dimensions for lumber. They know the thickness will be somewhat less, but the width should be exact.
However, if they do specify 1" NET thickness, then yes, you have to start with 5/4 (1 1/4") lumber and dress it down to 1" which would be required to get the necessary strength for that piece.
Often if you go to a real lumber yard as opposed to one of the big box retailers and you ask for some 1 x 8, they will say "do you want nominal or do you need at least 8" wide".
Read any notes that go along with your plans (fastening schedule etc). The designer should have notes on what is and isn't acceptable with regards to the dimension of the lumber...
Yes, Plywood is "real" wood
A "professional" is someone who gets paid for their work - it doesn't necessarily mean they are good at it