I agree it's very oddly written (note to Hunter et al: High Guard had simple formulas for everything. Why were they left out of this book? The result is far more confusing than HG ever was, even with the same desired end results.)
The problem here with the armor rules is that they never explicitly state that you have to multiply by the "Armor Factoring" number that you recorded when you designed the hull (pp 258-9). And they keep confusing Armor units with tons. Every instance of "units" in the example should read "tons."
Once you make that correction, the example appears to be correct. A 200-ton ship has an "armor factoring" of two. At TL 12 each unit of armor weighs two tons. The first point of armor weighs twice as much as later units. Thus, Armor Rating 1 requires 2 * 2 * 2 = 8 tons. Subsequent Armor rating points weigh 2 * 2 = 4 tons each. So Armor 12 weighs 8 + (4 x 11)= 52 tons.
I can't help with the System Defense Boat -- it's simply wrong. At TL 14, a 200 tons ship would require 30 tons of armor for AR 14. Rating 1 = 2 *1*2 = 4 tons. Subsequent factors weigh 2 tons each. Total is 4+ (2 x 13) = 30.
Fortunately, the SDB can simply take the missing displacement out of cargo, reducing cargo to 3.6 tons and increasing cost by MCr1.5.
(FWIW, the crew requirements are wrong too: by the rules, this ship needs 1 pilot, 2 engineers, 2 gunners and a medic.)