It sounds like you are talking about a solid. Solid stains are VERY difficult to remove. Nothing you can buy on the homeowner market is going to do the trick. Most professionals wouldn't even try to take on the job if they are experienced. You may have to resort to sanding it all off or just priming it and recoating with another solid like Cabot's or Deckscapes.

Can someone tell me how a stain can be a solid?
To me if it's a solid, it's paint.

All the 'stains' are paint, they just have varying amounts of pigment.

Solid stains are about as close to paint as you can get except the surface film is more breathable than paint. Pigmentation level is not what separates a solid satin from a penetrating one. That is too broad of a statement.

All the 'stains' you are using are paint formulations. They contain more or less solid pigment (that cannot penetrate anything) and differing levels of binder (some of which can penetrate depending on formula).
The pigment sits on top of the wood and is held in place by the binder. If there is enough binder present it mauy penetrate into the wood before hardening, but if you section a pices of wood after applying the finish you can typically see how shallow the penetration actually is.
Even aniline dye (a true penetrating 'stain') rarely goes much more than 1/16 of an inch in softwood without being driven in under pressure.

Now your answer is more complete, brick, but still not 100% accurate. I have seen Ready Seal (a paraffinic oil based sealer with no curing agents) penetrate halfway through a 2"x6" piece of cedar (as viewed via cross section).

That is a selaer, not a stain by any definition.
It is not the same.

Therein lies the discrepency. People use the words interchangeably. Other than a "solid", noone uses stain on an exterior wood job. Tinted wood sealers are the norm.

When someone says "stain" I automatically assume they are refering to a sealer unless they are talking about a solid. In which case yes, we are talking about watered down paint that forms a film and does not penetrate the wood.