Your query of safe and expensive but price limit of $2,000 is confusing.
Irregardless of that confusion, there is no confusion about the Fiat - stay away!
First of all, I'm gonna assume you meant safe & inexpensive since your limit is $2,000 (or was that a typo?)
As Trout53 said that is definitely "beater" territory and your not likely find any with less than 20,000 miles. But the ones I mention below can be found with less than a 100,000 miles and are capable of going 200,000 without major repairs. If you live in the rust belt, cars over 10-15 yrs old will be rather rotted. However in other parts of the country, there are a few sleeper models out there that can be 15 yrs old and still have a reasonable degree life, economy and reliablitity. To be safest, don't buy anything without air bags - most cars after 1992 came with them. You may have to hunt for them but they are out there.
1. Ford Escort. From 94 to 2003 the Escort use a lot of Mazda part and was generally underappreciated . Although somewhat tinny, they are both relatively reliable and economical. If you can find a lightly used one owned by a senior citizen it could be a great car
2. Mazda Protege: Similar to Escort a bit more upscale and better handling.
3. Geo Prism: This is a Toyota Corolla in disguise and a good bet.
4. Honda Civic. The 1992-97 Civic was a great little car and there a still a few around that haven't been bought by teenagers and riced up.
5. Generally I'd avoid Chrysler products with the exception of the PT Cruiser. Quirky looking, these mini-hearses were actually very comfortble, roomy, and relatively reliable. Fuel economy is so so. Again lots of them out there, look for low mileage model owned someone over 40.
Now, if you are looking for $20K cars, the options are wide open! Any Mazda2 or 3, Honda Civic, Toyota Corolla, or smaller Kia / Huyndai will get you where you want to go safely, economically & be reliable. Since you can drive a stick, you could even get a new one of the above for $20K.
Best wishes and let us know how the hunt went!