Rechargeable flashlights come in two varieties: Built-in and Removable.
Built-in rechargeable flashlights have batteries that cannot be removed from the flashlight. The main advantage of this type of light is the added protection to the electrical components from water, making the flashlight better sealed which increases the waterproof rating allowing the flashlight to work at greater depths underwater. Replaceable battery flashlights can still achieve the same IP waterproof ratings as built-in batteries.
Flashlights that come with replaceable batteries have the advantage over built-in batteries because the lifespan of the flashlight is not dependent on the longevity of the battery. A new battery can be swapped for the old and the flashlight should have the same output as the day it was manufactured. Current Li-ion batteries have recharge cycles of around 500, so removable battery flashlights are ideal for people who plan on using their flashlight frequently. Some require custom batteries which keep the customer tied to the manufacturer, however battery replacement is generally so infrequent that this relationship to the brand is not a major downside, especially if the custom battery is tailored to the equipment.
Any device that uses a replaceable battery is technically rechargeable if the battery being used is also rechargeable. This makes purchasing a rechargeable flashlight a little confusing because some flashlights can charge the battery via a port on the flashlight and some cannot. The newest flashlights have a USB-C port built into the flashlight and are fast charging capable. If there is no port built-in to the flashlight, the batteries must be charged separately.