Start by looking at the diagrams at
https://en.wikipedia.org/wiki/Coaxial_power_connector.
Many equipment-side sockets for this type of connector has three connections: one for the inner contact, one for the outer contact, and one for a "normally closed contact", which makes contact with the outer contact when no plug is present, but disconnects when a plug is inserted.
(What actually happens is that the plug has a curved spring contact that touches the outer metal on the plug. As the plug is pushed into the socket, this spring contact is forced outwards. This disconnects the spring contact from the normally closed contact; normally, when no plug is present, the spring contact pushes against the normally closed contact, but not when it's pushed away from the normally open contact by the barrel of the plug.)
In the design in the video, the normally closed contact provides the battery voltage to the circuitry. When no jack is present, the normally closed contact will supply the circuitry from the battery, but when a plug is inserted, the normally closed contact will disconnect, and the circuitry will lose power.
In his case, he was switching the positive side of the battery, so he was forced to put the positive polarity to the outer contact, because that's the contact that the normally closed contact touches.
This convention, positive to outer contact, is actually the original way that this power connector was wired, back in the dim dark days of early transistor radios and cassette players, on which it was originally and widely used. The switching operation was different then, because the power adapter was used for powering the unit instead of using the batteries, which were not rechargeable in those days. So the normally closed contact connected to battery positive, and the outer contact was the positive supply to the circuitry. But I digress
If you want to use the normal polarity, i.e. negative to outer contact, you would have to switch the negative terminal of the battery and circuitry. The inner contact would connect to the circuitry positive and the battery positive. The outer contact would connect to the battery negative. The normally closed contact would connect to the circuitry negative. Provided that the charger is isolated, and you use a plastic recessed socket, this should be workable.