I'll echo some earlier comments:
Although the parts store diagnosis said your cables are good, it is a well known fact the standard cable (Fiat factory size) from the alt to the starter is not sufficient. A heavier gauge cable is necessary for proper function. That is an excellent upgrade regardless of all other issues - and it might solve your problem. To almost the same extent is the cable from the battery to the starter. That connects the other half of the charging system together. So upgrading it is also a great idea (although it is a long one and a bit more difficult to do).
Proper tension on the belt is necessary. The fact it was loose suggests there may be other minor maintenance issues??? Not suggesting anything on your part. But things like all of the connections do become contaminated and highly affect the charging system's function. Remove, thoroughly clean with a wire brush and solvant, then tighten all cable ends, all starter wires, all alternator wires, all of the ground connections, etc.
The indicator light and charge gauge on the dash are not reliable sources of the system's health. Unfortunately this is another of the many weak areas on Fiats. Do not go by what they say. Check all of the charge levels with a decent multimeter at the various locations as suggested already.
IF (I'll repeat) if all of this is done properly and the tests with a multimeter still indicate you have insufficient charging, then - and only then - consider replacing the alternator. At that point you can weigh the cost options. I know what you mean about poor reviews for chain store rebuilt units. But if your budget is limited it may be a decent option. Worst case scenario you get your money back and do something else. Personally I've used a couple of them in the past with excellent results. But I can't assure you if that will happen again or not. Also look for another rebuilder to see what price they offer. Even if it is the same as other options, in my opinion it is a better choice. But all of this should only be after you have thoroughly exhausted all other possibilities. Again, no offense intended, but if you just now discovered the fan belt wasn't working then I don't think you have properly diagnosed the problem and could be throwing away your money by buying replacement parts. I mean all of this with good intentions to help you.