I had to read this several times before I realized that we're saying more-or-less the same thing.
If it's powered, it's an amplifier. I can't think of any use for a one-way booster amplifier in a modern cable system.
Anyway, from a layman's perspective, a splitter is like a tee fitting in plumbing. It's how you add stuff on, wherever it's convenient to do so.
No guarantee that the cable company's internal IT systems can accommodate this. You'd have to check in with them. In any event, it would be silly to pay the cable company twice for Internet service at the same address. Not a practical solution.
Meaning either the MOCA network adapters I mentioned earlier, or G.hn coax adapters. Doesn't matter which. Just two different standards developed by different engineering committees solving the same problem in different ways and flinging poo at each other. That, and different companies collecting the patent royalties. I mentioned MOCA first because it's what Verizon has been using, and I happen to have been peripherally involved with it.
You definitely need a Cat5 cable to connect your router to the adapter. From there, the easiest thing to do is connect the adapter to the house coax network, using a splitter. You would also connect the coax coming from the shop to the house coax network at the most convenient point, again using a splitter. Now, the shop coax is a part of the house network. Not only does this simplify your wiring, it also lets you put a TV in the shop by simply renting another set-top box from your cable company.
Speaking of which... if you're renting the router from the cable company, make sure that it doesn't have any labeling to the effect of "MOCA 2.0 compliant" or "MOCA Alliance blah, blah". If it does, you might get away without the external adapter.
Typo there... he meant RG-11, which is the most common kind of coax used for indoor cable installations. In any event, you have to figure out what is going to plug in to the Ethernet side of the adapter. If you're going to be using a tablet/smartphone or laptop, you'd want a WiFi access point. For a desktop or streaming device, Ethernet over Cat5 is faster and more reliable. You can also do both.
If you do decide to go wireless, you can buy a WiFi access point with built-in MOCA (or G.hn connections), so you don't need a separate adapter. I think I saw something like that on the Actiontec website (and I'm sure there are others).
A cost-saving option if you (or a friend) happen to have a WiFi-equipped router lying around. Unless you (or a friend) have some networking know-how, you'll pay in aggravation what you're saving in equipment purchases.
I think that is where TPG was going, although it wasn't clear to me from either of his posts (maybe they escaped from the PUI thread? ). The idea is to use a wireless connection rather than the buried coax. This can be done with pro-sumer equipment, but to make a long story short, it's not worth it. More complex, more expensive, more engineering knowledge needed, more installation work, and less reliable/slower connections. The only good reason to consider this is if you find that the buried coax is broken and you can't fix it.
Can't quibble with that!