Small Cabin

I wouldnt to either. I would put the 1in foam directly on the logs and screw furring strips through the foam into the logs.

We are doing something similar to our new home. 2" pinkboard on the inside. The ridgid insulation will make a good thermal break. I will install 1x3 lathe boards for nailers installed over the insulation where the studs fall for nailers.
May I suggest putting the topcoat on the TnG before installing so that back gets sealed also . Will help with wood movement.
That will look real nice

No need to use studs in between the foam sheets as you show. Fit the foam sheets against the inside of the wall, butt the edges and tape if you want the best air tightness. There are round plastic washers that can be used with screws to hold the panels in place while you work.

As was suggested you can use 1x4 boards secured vertically over the foam with screws that go thru to the studs. Then fasten the T&G to that in the usual manner with nailing thru the tongues.

Staining and applying at least one coat of clearcoat before installing the T&G is a great idea. Sometimes when T&G is stained or painted after installation and the wood dries out more there can be color gaps in the v-joints of the T&G.

It may be worth your while to go to 2" foam. ???
Two layers with the seams staggered make for better air sealing but may increase the cost.

For the furring strips - do you do every 16 in ? Also is 1in enough you think

The wall are 6x8 logs so there are not studs.

I'm curious... Why do you want to cover them?

The more insulation the better I guess. Just wondering about a vapor barrier. Don't want moisture trapped in the walls. The insulation will be a moisture retarder. If the seams are taped it should keep most moisture away from the logs and hold out any air leaks better. Also open n closed cell ridgid insulation. My preference would be the closed cell, pink or blue. Open cell such as the white bead stuff can attract mold easier also. You can purchase 1x3 lathe fairy inexpensive. Nail into the logs. Maybe measure out where the log is widest to get best nailing. The plastic cap nails should be fine. Good luck n have fun

Are you saying that moisture would accumulate behind the rigid foam ?

Is this for full-time use or part-time?

I was hoping not to use vapor barrier as I thought the rigid foam board would be the actual vapor barrier that goes directly into the logs then strap it for T&G.

I'm using timberlock screws. They are construction grade and will not bend 4" would work. Less holes is better. Try 2' between nails n check how sturdy it is.

Staples that long fired from what? Might work to handy-tack em down for initial fit but the f-strips do the hold down and provide a sorta leveled up nailer base for the wall boards.