You buy a refillable lighter once and top it up with butane gas after that. Rechargeable means the same thing here: a lighter with a valve at the bottom onto which you press a gas can.
The difference from disposable is not only the price. A refillable model is usually built more solidly, with metal instead of thin plastic.
A disposable lighter costs little, but you buy a handful a year and all of them end up in the bin. With a refillable model you only replace the gas. Topping up takes a few seconds and it lasts for years, as long as you fill it with clean gas.
This category holds refillable lighters from small and compact to heavier models that stay upright. With one, two, three or four flames, with a jet flame or a soft flame, singly and in value sets.
If you are after the gas as well, you will find it at lighter gas. If it mainly has to keep burning outdoors, have a look at the storm lighters.
When comparing, look at the tank size, the type of ignition, whether the flame is adjustable and whether it has a lock. If a filled lighter still refuses, the answer is usually in lighter not working any more.
If you do not know which model, start with how often you use it. Daily use calls for a bigger tank and a sturdy ignition, lighting the odd candle does not. Only then look at the type of flame.