Ptolemy gives Betelgeuse the nature of Mars and Mercury, and the Mars half is a direct reading of the colour. This is the same move Ptolemy makes with Aldebaran and Antares: the visibly red stars are martial.
That reasoning deserves an assessment rather than either endorsement or dismissal. It is signature reasoning, of exactly the kind found in the plant material, the appearance is read as a statement of character. But it differs in one respect worth noting: the colour Ptolemy recorded is physically real and correctly observed. Betelgeuse is red because it is a cool supergiant, and its spectral type confirms what the eye reports. Ptolemy's observation is sound; only the inference from it is arbitrary. Compare Sirius, which Ptolemy also describes as red and which is not, and where the observation itself fails.
The Mercury half supplies the inventiveness and quickness in Robson's delineation, and it tempers what would otherwise be a purely martial reading. Betelgeuse is generally read as fortunate (martial honour, preferment, wealth, enduring vitality), which distinguishes it from most Mars-flavoured stars.
Betelgeuse is not among Agrippa's Behenian fifteen, despite being one of the brightest stars in the sky. Neither is Rigel. The Behenian list is not a list of the brightest stars and was not selected on that basis; it is a talismanic working set, and its selection criteria are not stated by Agrippa, who attributes the whole scheme to Hermes Trismegistus, an attribution that means the true origin is unknown.
The name's derivation is uncertain, and the common gloss the hand of the giant rests on a misreading of the Arabic that entered European usage early and stuck.