It is noted in the link you mentioned:
Depending on your device and its configuration, you may have options for wiping internal storage, external storage, sd-ext, android_secure, and/or an option for formatting data. There’s almost no reason that you would ever need to use these items.
(Emphasis mine)
For a clean wipe, you can get rid of everything under /data except of /data/media/ which in particular gets mounted as the emulated internal storage by Android. In short, it houses your files and serves as the internal SD card (/sdcard). You can however get rid of /data/media/0/Android directory since it contains files related to applications installed in your currently installed ROM. Those files may or may not interfere in smooth functioning of your new ROM after a clean wipe.
Advanced wipe can be performed from TWRP through Wipe → Advanced Wipe → choose appropriate boxes. Remember, Internal Storage there means /data/media and Data means /data excluding /data/media/.
If you still intend to wipe everything:
As Salesman wrote, you can use an external SD card to store the ZIP. If your device (OnePlus One does) supports USB OTG then you can use a USB drive as well to store the ZIP. Otherwise, as Death noted, side-loading is also a good option. In the side loading mode, the zip is served or as you said, streamed. You would need version 1.0.32 or above of ADB client in your PC to do side loading.
To initiate side loading, from TWRP go into Advanced → ADB sideload → swipe to begin sideload. In the PC, provided that you've adb setup and device plugged in using USB, issue the command:
adb sideload PATH
# PATH is file path of your ROM's zip in your PC.
Once side loading completes, you can do whatever you want in TWRP.
By the way, using MultiROM for OnePlus One you can get rid of all of this dilemma.