Same as per Blue Skies.
When I was in 8°N and I had to explain the stars under a starry night to the peoples, I referred to the Southern Pointers as Alpha and Beta Centauri, and only after the explaination I would say "they have also a proper name...". This is because their proper names, Rigil Kentaurus and Hadar, are a bit less known than their Bayer designation. I checked it on many books and atlases. On the other hand, many bright stars like Sirius, Canopus, Achernar, Antares, Vega, Capella, and so on, are best known with their proper name, and rarely (only on certain academic publications) I saw them called Alpha Canis Majoris, Alpha Carinae, Alpha Eridani, Alpha Scorpii, Alpha Lyrae, Alpha Aurigae, and so on.
Concluding, both the names are correct, it is only your discretion using one instead of the other.
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PS: about the precession. Here in Sardinia we have an ancient megalithic monument aligned with the Southern Pointers. 3000 years ago, they were visible from my island; now these two stars are at least 10° below the horizon.
