they make a masi speciale LTD drop version, albeit with a different color scheme.
http://www.masibikes.com/fixed/speciale-fixed-ltd-drop/
anyway, my thoughts: just fix your current bike. you haven't mentioned anything about it, so who knows, but if everything else is in working order, no reason not to do it. Also, if you snapped the chain, it's likely because the damn thing was on there too long. If you don't take care of your bike, it won't matter if it's old or new, these kind of problems will arise.
if you do want to get a new bike, I'd recommend looking around more. The district is way too commuter-y (unless that's what you're into), and as aforementioned, if you really want to get into bikes (or at least want to keep the option open), it won't leave you much room to progress/upgrade stuff on your drivetrain.
The speciale is nice, but only decently so. Cranks and wheels are good, but everything else just says "ok" /pretty standard stock parts. Furthermore, you don't really need upgraded cranks right off the bat, especially if you're a new rider. At any point as a rider, having a resilient wheelset is very nice, but you can get a set of dp18s, or deep v's laced to formulas over at velomine, and stick em on whatever bike you want.
For different hobbies, the idea of "should i get entry level stuff and later upgrade, or get nice stuff from the get-go" really varies, but when it comes to bikes, I'm a big fan of going for something very adequate, but inexpensive. Maybe a bianchi pista?
http://www.bianchiusa.com/bikes/urba...ar/pista-blue/ or if you really want to go inexpensive, anything from bikes direct.
aside from the street cred you gain or lose from any of the cheaper models, the frames are pretty much all 4130 cromo (or reynolds 512. same difference. Also, even if it's only listed as "Cromo", it's most likely 4130. If it were nicer, it'd be a selling point), and pretty much the same "road-ish/track-ish" geometry.
when it comes to components, you'd be surprised how much mileage you can get out of the stock "crappy" parts. Granted, they're definitely cheap parts, but by the time you actually want to upgrade, or finally start to break parts, you should have a much better idea of how you want your bike to feel, and how much you're willing to spend to make it feel that way.