I'm really worried about professionalism.
My site would be something like
http://mywebsite.uy instead of
What do you think about this practice? What should I have in mind at the moment of choosing one?
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Get both domains and redirect
.uy. I agree with you that shorter is better, even though Uruguayans are used to the long version. The same goes for
.uk. While objectively
.uk is the better version, it only came out this year making it unfamiliar to users.
It entirely depends on your country. If it officially has two-level TLDs, like for example the UK and Australia, then a second-level TLD is absolutely fine -- no one thinks less of a .co.uk website. If it doesn't, and some third-party has registered com.<country code> and is selling subdomains then you shouldn't touch it with a bargepole, as there are many problems with that approach.
For your specific example, Uruguay has official *.com.uy domains, and so you'll be absolutely fine with one.
One of my main focuses when it comes to websites is "user friendly". (The other is security.)
With that in mind, I prefer to stick with "shorter is better" for two reasons.
1) It's easier to remember. It's easier to "say" when someone is talking to their friends or colleagues.
2) Less typing (it's shorter)
I know #2 may sound trivial (we're only talking about 4 keystrokes), but every little bit helps.
The United States only had .us for years. It was only when .us names started running low that they introduced .us.com (early 2000's).
Summary... I would stick with .uy instead of .com.uy