Essentially, no. While Thunderbolt 1 & 2 can power downstream devices, the maximum amount of power is not that much, and there does not appear to be any provision for powering upstream devices including the host computer. Macs can certainly not be powered/charged via a Thunderbolt 1 or 2 port. Thunderbolt 2-to-3 adapters do not change this and do not retroactively enable more power delivery on the Thunderbolt 2 side.
The Apple Thunderbolt display actually had a MagSafe connector in addition to its Thunderbolt cable in order to power a connected MacBook.
Else I'll buy one with USB-C to power the next laptop I buy, and use a USB-C to Thunderbolt 2 cable/adapter for now.
If I understand you correctly, your plan is to buy a display with USB-C as the display signal input and power delivery, and connect it to the Thunderbolt 1 or 2 port on a Macbook?
This won't work. There are reports that you can do this with Thunderbolt 3 based displays (e.g. LG Ultrafine 5K but note your MacBook Air does not support 5K), but it most definitely will not work with a display that simply uses USB-C with its DisplayPort alternate mode. It's not clear exactly which adapter you were planning to use but:
- Simple USB-C to (Mini) DisplayPort cables can only be run in one direction: USB-C on the host (computer) to a DisplayPort on the display.
- Thunderbolt 3 to Thunderbolt 2 adapters such as Apple's only work for transmitting Thunderbolt signals, not USB or USB-C's DisplayPort alternate mode.
If you want to buy a display that will power a future USB-C based laptop, choose a model where the USB-C port/cable with power delivery is merely one of multiple ports. You can then use either an HDMI or DisplayPort input on the display to connect to your current MacBook Air.
Note that the 2013 MacBook Air only supports DisplayPort 1.1 and HDMI 1.2, which limits the maximum resolution and refresh rate:
- DisplayPort 1.1 supports up to 2560x1600 at 60Hz; depending on the specific Intel GPU and driver you may be able to drive a 4K display at 30Hz. (From personal experience, the early 2013 MacBook Pro and the 2012 Mac Mini have trouble with this; during boot and in macOS 10.12 and earlier, there is image corruption. Once booted up, 10.13 and newer seem to work ok at 4K @ 30Hz. The 2013 Air has a slightly newer CPU/GPU, so I'm not sure how it behaves.)
- HDMI 1.2 (via Mini DisplayPort to HDMI cable on a MacBook Air) only supports up to 1920x1200 at 60Hz.