You're implying "not Rebuild" is ever really an option...
Seriously though. Rebuild is going to win on most boards, and I think this is one of them. Cards like Farming Village are tailor made to play in a Rebuild deck. You have the rare opportunity to Rebuild into Province with a card that isn't Duchy (I would argue you *always* buy Fairgrounds on 6 against someone who isn't playing Rebuild). Farming Village is good in a Rebuild deck (keeps VP in the deck), Cutpurse is great for hitting $5 and stopping your opponent from doing so, Lookout can rid you of a few Coppers, and the Rebuild player is going to be able to Tournament faster.
A really good player could carefully make the engine win some of the time, but I wouldn't take the engine bait here. I'm kinda lazy I guess.
I'm not sure if Rebuild wins on most boards. It probably does in practice since everyone (including me) always goes for it, but I'm really not convinced that it's the best strategy over 50% of the time.
You should ignore Farming Village, Fairgrounds, Cutpurse and Lookout entirely if you're going for Rebuild here against a non-mirror. Why? Because Fairgrounds costs $6 and you pretty much always want a Rebuild for $6. You have three Estates at the start, so Rebuilding them into Provinces gives you 18 VP already, and then you just need to Rebuild the Provinces into the remaining Provinces as fast as you can and then you win, basically. That should be your plan, at least. If things don't go as planned, you might want to get an additional Duchy or a Fairgrounds at some point if it starts looking like you need it, but there's Tournament which gives you the Duchy for free, so you still don't necessarily have to buy it. If you need the extra green but can't collide Tournaments with Provinces, it's still easy to purchase the extra green card on the fly when you need it, and the later you can get it, the later you should get it because that way you'll have the information required to make the decision about needing an extra green, and then you can actually do stuff with the Rebuild you got earlier instead of having a dead card cluttering up your hands. You should get a Province if you're lucky enough to hit $8, though.
Farming Village and Cutpurse are $4 cards and you pretty much always want a Tournament for $4 (I didn't do this in the second game and I'm not sure if it made any actual difference, but I'm pretty sure that it was a mistake). Farming Village isn't super good because you probably won't have too many terminals in your deck (if you're lucky, you'll get both Princess and Followers, but you don't need a Village for two terminals in a deck that doesn't draw itself every turn), and you probably won't have too many green cards in your deck either (because you only have 3 or a few more), so most likely it doesn't do anything for you. Cutpurse isn't super good because your opponent doesn't even need to hit $5 if he's not going for Rebuild, which is the only $5 card on the board. Tournament, on the other hand, is super good because having lots of them makes it more likely to collide them with Provinces, and it gives you economy while it doesn't slow down your cycling. You might want to do what SCSN suggested and open Courtyard/Silver instead of Tournament/Silver, I'm not sure which one of them is better but I personally like the Tournament opening more.
Lookout is horrible for Rebuild. In Rebuild games, you never have to hit more than $5, and Copper actually helps you hit $5 so it's not that bad. You still want to get rid of it if you can do so for free, because that way you'll draw Rebuilds more often, but you don't want to get rid of it bad enough to spend an early $3 buy on a card that takes up a card slot in your hand without bringing you closer to hitting $5. And you obviously don't want to trash Estates.
On the other hand, all four cards are very much useful for the engine player.
I think that the Rebuild strategy might be a bit stronger here, but it's pretty close.