I bit the bullet and bought 2 x highly regarded chargers with USB-PD Function, in the faint hope that it would work with the HP Spectre 13 (which has 15W 3Amp as its native charging).
The chargers I tried were the iVoler 75W USB Type C Charger with Power Delivery, 60W USB-C Port+15W 3 USB Ports and the Anker Premium 5-Port 60W USB Wall Charger PowerPort+ 5 USB-C with Power Delivery. Both of those are high quality products that do seem to work well with all my other USB-C devices.
In short, it does appear that the HP Spectre 13 does have some form of 'charging DRM' to restrict the usage of third party chargers. It does not appear to work with standard USB-C PD chargers of the same wattage / voltage spec. The iVoler and the Anker DO charge the laptop when the laptop is OFF, not while it is functioning. Furthermore, if plugged in while off, it will cease charging as soon as the laptop is booted up.