Smart to be cautious. I'm with you that the legending is not particularly clear for the end-user .
A word or three about that style of plug. Where a phone plug, due to the physical construction of the plug and the corresponding jack, virtually always MUST use the shaft/shank as ground, barrel-style plugs have a choice whether to use centre-pin negative or centre-pin positive. Back in the early days of calculators, when they needed to use external power supplies because of the current draw (my old Sinclair programmable used a neon-like display that was neither LCD or LED; almost nixie-like), many would use mini phone plugs on the wall warts. Where 2.1mm barrel plugs were used, some manufacturers used outside-neg and some outside-pos. I have piles of power adaptors with polarity switches built in.
Where a pedal has to switch between an internal battery and an external power-source, however, it is the shaft/outside that must push aside a contact to disconnect the battery and feed the power via the external source, if only because there is no tip, like found on phone plugs.
Where a device does not or can not operate via internal battery, it becomes the manufacturer's choice what plug polarity they wish to use, because there is no de-connection to be made, only a connection. And indeed, some older pedals used the opposite polarity. And some used AC supplies, indicated by two wavy lines near the jack, rather than two concentric circles. Where the pedal expects to receive an AC voltage, the manufacturer has chosen to do the regulation on board. In many instances, such pedals take a single AC voltage and provide a bipolar supply to the pedal (so, 12VAC turns into +/-15VDC), or perhaps the intent is to provide multiple supply voltages for different parts of the circuit (e.g., 12VDC for relays, 15VDC for audio, and 5VDC for digital circuitry).
All of that said, given how many pedals are designed to use a battery, even if a fresh 9v battery lasts only for 2hrs, the use of outside-pos/center-pin-neg has become fairly standard in pedals. It is to the point where even if there is no way in hell the pedal could use a battery and MUST be externally powered, the power jack uses the same orientation/polarity as battery/external pedals, simply because there is a reasonable expectation that the pedal will be powered by a shared supply, and not necessarily its own proprietary wallwart.