While
@sneefy is correct - tires are a regular source of postings on the SubaruForester.org site, this may be a bit of a different question being posed....
I'd suggest the Continental TerrainContact AT tire for you,
@2020forester ... they're not as "off-road" as some of the other makes and styles out there and seem to be a good blend of all-season and all-terrain abilities. I've got a set on my 2014 Forester and have no complaints (so far)... The mileage to a bit of a hit as the tires have a higher rolling resistance and a more open tread pattern than your typical all season but have also have some added grip for off-the-beaten-path excursions... You may also notice a bit more tire noise/road rumble from them - again owing to the more open and aggressive tread block designs. But again, they're certainly NOT as noisy as some of the other all terrain tires out there.
One of the benefits of an all terrain (vs all season) tire is a thicker bit of rubber (sidewall) that can withstand a bit more and is stronger. While this may not help with a nail in the tread, it does help when on the trails with rocks n debris. If you're going on an exceptionally rocky path, I'd also suggest investing in an air compressor/pump to partially deflate (and then re-inflate) the tires when on those rocky roads. Drop about 5 to 10 PSI (low to mid 20s) and that will also make the tire more flexible and able to take a bit more of a hit before puncture. Of course, this is all subjective and your mileage may vary.
Did the dealer ever find out what was wrong with tire #2? Just defective or ...?
And as
@Kean mentioned - keep an eye out on where you're driving and parking... if you go near construction sites, it may be a good idea to change your route (if possible) to avoid those areas if possible.