Since I was once working as salesman in an Authorized Reseller in Australia, let me enlighten the essential parts :
Apple had implicitly mentioned in the press release that the MC model would not have the same spec as its sibling, which means the MC model is not the same with the 3rd Gen model.
So what is a MC model ?
MC model is a 2nd Gen iPod with a modified bootrom so it was meant to be UN-jailbreakable. But as you see, maniacs out there were able to jailbreak it too (this really bugs me). You may notice that the MC and MB models are quite different but they were the same since both of them were A1288 iPods.
Since the time when Apple made the press release, many stores advertise the MC model as the 3rd Gen. This is absolutely wrong ! Although my store didn't notice about the differences until iOS 4 was released, we took the liberty of advertising it as The New iPod Touch 2G 8GB in February 2010. Actually, Apple ever advertised in fine text that iPod touch 8GB won't support all features on iOS 4 in March 2010. The salesman at your store should've told you this before.
About the method, well, beats me. Most stores have their own way to distinguish such as the part number (MCXXXX and MBXXXX), however since part numbers are different in every sales region, Apple used a more common way which is the model number as shown in http://support.apple.com/kb/HT1353#iPod_touch_late2009. However, you won't be able to check the model number unless you buy the iPod, opened the case, and use a magnifier on its backpanel. A lot of online websites made profit from this thing, since people bought the product and they complained 6-12 months later (assuming the purchase was done in late 2009 - early 2010), and they were terribly dissapointed.
If you carefully notice that the link in the webpage for the 4th Gen iPods was HT1353#iPod_4G while for the 3rd Gen iPods was HT1353#iPod_touch_late2009. See Apple designated the iPod touch which was released in 09/2009 (at the same time with the press release) only in 32GB & 64GB capacity. So what is the designation for the 8GB ones ?
They were just iPod Touch 2G 8GB - MC model, any other name for this iPod would be considered unlikely to be accurate.
The store where I worked once also experienced the same problem back in May 2011 where users who bought the product before our new advertising was launched. Well, we apologized because it was my store's mistake and told the customers why wouldn't they bring the problem back to us earlier ?. Well, that's not the important thing (as long as the customers were happy, it was enough for me). The customers demand the true iPod touch 3G 8GB in exchange for their old The New iPod Touch 2G 8GB. But there was no such thing called iPod Touch 3G 8GB, so we made a deal with them. We let them keep their iPods and offered them the new iPod 4G 8GB for only A$49 (about 20% of normal price at that time). since there wasn't a 3G 8GB. Fortunately, there were only 3 customers at the time and we revoke one of them because the iPod unit wasn't functional. They accepted the deal and we sold 2 iPod Touch 4G 8GB for A$98.
Now since you would likely to demand compensation from your local Authorised reseller, tell them how we settle things here in Australia. Your local store might've misled you when buying your product, so you have the right to demand compensation even it's been 2,5 years since the released of the iPod touch 3G. I think that refund might be an overdo since it's been quite some time, so I suggest you demand for 80% discount for purchasing the new iPod touch too :).
Best Answer
Did you sync your iPod Touch with iTunes recently? An iOS device won't download newer versions of an app if they can no longer run on your current iOS version, but iTunes doesn't stick to that rule. So if the Facebook app was updated in iTunes, and then you synced with your iPod Touch, it may have been updated to a version that it can't run, hence the immediate quitting.
You can verify this by going to Settings → General → Usage and tap on the app under the storage setting. You'll see a version number, and you can Google that to see if you have a version that supports old iOS or not.