You've answered your own question. It is something in the bulb socket or the dual female connection on the inside the headlight bucket. It is an easy trace, as the high beam indicator circuit is a solid dark blue wire off that dual female connector where the solid blue wire for the high beam headlight bulb circuit is connected.
There maybe some corrosion/damage inside the bulb socket itself, or with the male/female connection inside the headlight wiring.
If your high beam headlight is good and bright, it is fed 12V off the same dual female socket noted above, so the issue is likely in the two areas I have previously outlined.
Oh there is also a ground wire on the indicator bulb circuit that could also not be grounding from corrosion, so check both wires off the indicator bulb connection.