The second house is a post-ascension, an epanaphora, which the Latin tradition calls succedent: it follows a pivot and is held to act with less immediate force than the angle before it. Its Hellenistic name is the Gate of Hades.
The name has a structural reason behind it. Counting by whole signs, the second place stands at no classical aspect to the first: not a sextile, not a square, not a trine and not an opposition. It is in aversion to the rising sign, unable to see it, and the four places in aversion to the Horoskopos are the second, sixth, eighth and twelfth, which are also the four the tradition names as difficult. The judgement follows from the geometry rather than being imposed on it.
On the cusp, the systems part company sharply. Whole-sign puts the second house in the second sign from the ascendant, whatever the degree. Equal puts it exactly thirty degrees after the ascendant. The quadrant systems, Porphyry, Alcabitius, Campanus, Regiomontanus, Placidus and Koch, each divide the arc between the ascendant and the lower meridian by a different rule, and their second cusps can lie many degrees apart in the same chart. The disagreement widens with latitude and is already substantial in northern Europe.
The practical consequence is that a planet reported in the second house has been reported in a system, and changing the system may well move it. Where the question is money and possessions, that is not a small matter.