Racing tire damage and a training tire replacement

I was on the fourth weekend ride with my new bike, the first nice one I've owned. It has Bontrager X-Lite wheels and (unfortunately) Bontrager X-Lite tires. I had them at 110lb, the recommended max is 120. I ride over not-so-great roads, bits of glass and ruts are pretty common. I didn't see any debris or feel the tire hit anything, but the rear went flat in a second with a 3/4" gash on the sidewall. I had everything needed to change it, so no problem. The kevlar clincher just about jumped off the rim. I got my spare tube on and started to pump it up. At around 40lb there was a loud pop and I now had twin ruptured tubes. Three and a half miles in DMT shoes with Time cleats is a great way to stretch those hamstrings (and chew up cleats).

This is the first bike I've owned with a racing tire.
1. Did the second tube blow because the tire wasn't rigid enough to support the tube at pressure?
2. Are <250gm tires just that vulnerable?
3. I'm used to buying 400gm commuters that last 3000mi. Is there a 250 to 300gm road trainer in the $20 to $25 range that would last 1000 to 1500mi?
Thanks.

Glass cuts can require "booting" to prevent the tube from extruding out of a cut. How big of a cut do you need to boot? Anything over 1/16" will benefit from a boot.

What to use for booting? Park Tool makes a self-stick boot. Or you could cut the beads off an old tire and cut sections off for boots. On the road, you can use a dollar bill, folded many times or you can use a PowerBar Wrapper. Whatever you use, it should not stretch but it needs to flex. Glue isn't required, it only makes it easier to keep in place.

1. The casing of the tire was cut through, allowing the tube to push out and blow. As weissluft said, you can carry a boot for roadside repair of these kind of cuts. Some people use a folded dollar bill, or scrap of old worn tire.

2. Some <250 gram tires are much better than others. Racing tires are made for racing, not for bad roads with glass. EG, I've had lots of cut throughs with Michelin race slicks, none with Conti GP 3000s.

3. I'd forget your price limit and weight here and buy the best tires for your needs. A $50 tire that lasts for 3000 miles on the rear can be a much smarter choice than a $20 bargain that only lasts you a few rides before it cuts through, not to mention the better ride, handling and grip. Besides, only the rear tire really wears.

I get 2500-3000 miles on Conti GP3000s with no cut through flats; these are under 250 grams but may not be best for your dirty roads. Suggest you try Conti UltraGatorskins, Specialized Armadillo, or another tire with a strong casing that's resistant to cut through. Lightweight "race" wheels and tires have lots of appeal when we're shopping, but it's nothing compared to durability out on the real roads.

Thanks for the tip on using a boot.
And you're right, the objective defines the solution. I'm better off with a damage resistant tire that may cost more than a fast rolling light weight that turns a ride into a hike.