The way it works is that each player "has" some set of cards. If you automatch, the cards are selected from the combined set of cards of each player.
So if you (I assume you have no subscription) are matched with a player that also has no subscription, you two together "have" just the cards from the base set, so your game will consist of only base cards.
If you are matched with me (I have all the cards), the game will consist of a Kingdom from all existing cards, and it can include those banner cards (usually card "landscapes" due to the picture format).
At the moment it is, for a player without subscription, only possible to restrict themselves to base cards, or to whatever there opponent has.
So your example, playing only base + 1 expansion isn't possible.
We are aware of the problem and there is an overhaul of the client in the works.
Among other things, it introduces card pool levels from 1 to 10. The card pool for level 1 is just the base cards, the card pool for level 10 is all the existing cards, and the level's in beetween gradually include more cards, based on concepts. So while landscape cards exist in many expansions, they are introduced only in the card pool for level 7, and then only one kind, the Events. Other kinds of landscape come later.
For rated play, you can then specify: "I want to play at most with cards from card pool level 2."
This way you are not overwhelmed with new cards. As you learn those cards more, you can then increase the card pool level you are comfortable with, at your own pace of playing/learning.
In addition, it will be possible to "pretend" you only have a subset of the expansions you actually are subscribed to. That means you can say "I only have base and Intrigue", so that is your contribution to the card set for a given game. That doesn't prevent your opponent from having more cards, but it helps when you play against a bot, for example.
Together those 2 features should help players like you to ease into the game at a comfortable pace.
Unfortunately, it is unclear when the overhaul will become public, as there are other areas where work needs to be done.