I am surprised that oil based paint is still available in the US as for sheen I have no idea because real shiplap would absorb all the paint you would slap on it my guess is you have some smooth boards and the sheen can be whatever you like and all sheens are available in latex or even better acrylic paint that won’t stink up the house for weeks.

IMO, just use a flat or eggshell finish with a good quality latex paint.

You encouraged me to do a little research because you confuse your paint terms in your post. There are in the main two types of paint on the shelves for the homeowner: oil-based and latex.

There are four basic ingredients in paint: carriers, binders, pigments and additives. Carriers usually make up the largest percentage of volume in a can of paint and transport the essential ingredients from the can, to the brush or roller and onto your wall. Water is the carrier in latex paints and mineral spirits is usually the carrier in oil-based paints.

Binders are the expensive part of the paint can and the essential element to a paint's performance on your wall. Latex paint binders are often acrylic, vinyl acrylic or styrene resin. Binders are also responsible for a paint's sheen. 100% acrylic paint is typically a better paint (but not always) than vinyl acrylic, which is often used in cheaper paints.

Pigments give paint its color and Additives provide enhancements for workability, performance and other benefits like mold deterrence.

Flat paints have a higher concentration of pigment to volume than gloss paints and because binders are usually an expensive component, this is why gloss paints are usually more expensive than flat paints. Flat paints will hide dimples, dents, saw marks and other imperfections in the shiplap better than gloss paints because of the greater pigment coverage and less attention-grabbing reflections from high and low points.

There's really no need to use oil-based paints these days. Latex paints have progressed in technology to the point where arguments against bonding capability, uniformity, pigments, wear, life expectancy, etc., in comparison to oil based paints are all moot and the traditional advantages with cleanup and ease of use are still there. Plus you don't get high for a week or two after application.

Ship-lap?!??! Oh my.... haven't heard that term for a while!! YEEEEARS ago, most paint companies were phasing out most/sometimes ALL oil based paints. ACE was dropping most oils about 8 years ago.

I hardly EVER have to tint full oils anymore! I've been in retail paint, etc., over 15 years. Besides, the Glycol-based colorants used to tint both oils & latexes for 30+ years, have largely been phased-out too. In 2013, ACE, C2, Valspar, & others, started phasing in their new Low/no VOC Acrylic colorants. This change often required new tinters, or at least a lengthy cleanout of existing tinters. A few stores, maybe kept old manual tinters around, if they could still get the glycol colorants. Then, there's also the issue of tinting software!!

>>> And....you can't tint OILS with ACRYLIC colorants!!!

Faron

Forget the oil, if you can even get it .Eggshell or semi gloss for the sheen.

Fine paints of Europe Hollandlac satin will deliver the look you want.

There is an argument for using an oil or shellac based primer: they are generally superior in sealing the wood and not raising the grain, as do water based products. An oil enamel undercoater, after a light sanding, will be super smooth. Either an oil or water based finish coat can be painted over it. Further, should the wood have water soluble tannins in them, the tannin will not leach trough causing yellowish or reddish stains. Most evergreen source woods have tannins, and so does red oak. Cedar was always primed with an oil primer for this reason. Knots are tannins on steroids! Should this be old ship lap which has been exposed to nicotine over the years, again oil or shellac primer will stop it from bleeding through.