The notions of Bayes Correlated Equilibrium and Bayesian Nash Equilibrium in a game with one player
The terms experiment and information structure are used interchangeably by the literature, so both terms are valid regardless of $N$.
I agree if $N=1$ there is really no game, and it is rather a single-agent decision problem (or decision problem with uncertainty), and yes, you don't have a player, rather a decision maker.
For the same token, you don't want to say the solution concept if BNE, instead you want to say that the agent maximizes his expected utility given the information she received. 
I was not aware of the term "one-player BCE" but Bergemann and Morris coined the term BCE, so if that is how they call it, it is correct.
In terms of other references, this is an earlier version of that same paper. However, I think that the paper you linked is the right paper for citing the concept of BCE. 


In this later paper, the same two authors made the remarkable connection that the problem of Bayesian Persuasion popularized by Kamenica and Gentzkow could be interpreted in light of the concept of BCE when $N=1$. So perhaps, this later reference of BM can be useful if you want to explore more the case of $N=1$.