The teleporter in one of these gas processor bases has blue rings of light around it and is useable by me even though I don't have the rep for it. How did I manage to unlock it, and can I repeat the feat for other poi in range?
Teleporters have since long been changed to only work if both depart point and destination have a friendly (or better) reputation. Neutral is not enough. Last time the teleporters were different it was in version 1.2 or so. If still you are using a very old savegame started in 1.2 or less, then maybe you still have such an effect. If you delete your cache folder and make a file integrity check to correct game files then all should be as normal, meaning you should NOT be able to teleport from your screenshots provided that you are indeed neutral and not friendly; check this in the faction tab, and don't be fooled by some NPCs walking around, these may be from another faction that have nothing to do with the station faction. (the registry tab will tell you the exact faction for this POI).
During my plays I've met another gas refinery. It's like you say, but all teleporters work that way, you always can teleport to YOUR bases and friendly teleporter locations, however teleporting to neutral teleporters will not work. This means you must care not to have a 'one-way' ticket only and get trapped: while you can teleport from neutral bases to your base, you will NOT be able to return. Just remember not to teleport from neutral bases. I use to place a solar teleporter near to those stations which are neutral and have interesting traders inside, that's the way around the problem.
