Well it's all semantics of course but the only difference between a carpet and a rug is the size. A rug is simply a small carpet. Fitted or wall-to-wall carpets are a fairly modern thing and the expression was used to differentiate them from traditional carpets, which are not fitted, such as the carpet in the second picture above. When Aladdin got on his flying carpet it was neither fitted nor was it a 'flying rug'.
Now the tasteless and unseemly fitted carpet has become so common that young people such as yourselves don't even realise that proper carpets are neither fitted nor wall-to-wall and think that the only kind of carpet is a fitted carpet and everything else is a rug. Such is the paucity of your taste and discernment concerning floor coverings. Bah! Humbug! Etc.