There are three terminals on the headlight, High Beam, Low Beam, and 12V return. Since you said you rewired the headlight, I have to ask if you power both High and Low beam at the same time. This would be very bright but, make any lamp overheat and fail prematurely.
12-13 volts on the regulator is a normal voltage reading. However, you need to set conditions that would allow the voltage to be higher, such as a fully charged battery, and the alternator spinning at 4-5000 RPM. Then read how high the voltage goes. If higher than 14.5 V then the vreg or some part of the charging circuit is suspect.
Cheers,