Which inference can be best made about why the king is described as “semi-barbaric” rather than wholly barbaric?
The king is regarded as "semi-barbaric" in the sense that he has been influenced by his distant Latin neighbors. His ideas are described in the tale as "polished" and "sharpened" by Latin influence, which implies that he's steeped in the culture and learning of antiquity. However, despite this, he still retains vestiges of his indigenous heritage, whose barbaric nature manifests itself in the cruel "lady or tiger" punishment the king imposes upon those who displease him.
Yet there seem to be significant similarities between the "barbarism" of the king's own culture and the "civilized" life of the Romans. The king has a large arena in which brutal gladiatorial contests take place and in which the religiously unorthodox regularly meet a savage end at the claws and teeth of starving wild animals. This is precisely what used to happen in ancient Rome.
Had the king restricted himself to this form of punishment then perhaps he would've been regarded as fully civilized rather than in the least bit barbaric. The suspicion remains that his "lady or tiger" method of punishment is considered barbaric, not because it involves violence and bloodshed involved (after all, Roman punishments were often far worse), but because it is an expression of indigenous culture, which is not something imported from Rome and therefore deemed to be "uncivilized."