Are PostgreSQL clusters and servers the same thing? From the official documentation (emphasis mine):

Chapter 22. Managing Databases


  Every instance of a running PostgreSQL server manages one or more databases. Databases are therefore the topmost hierarchical level for organizing SQL objects ("database objects").


Chapter 22. Managing Databases -> 22.1 Overview


  More accurately, a database is a collection of schemas and the schemas contain the tables, functions, etc. So the full hierarchy is: server, database, schema, table (or some other kind of object, such as a function).


Chapter 18. Server Setup and Operation -> 18.2. Creating a Database Cluster


  A database cluster is a collection of databases that is managed by a single instance of a running database server.


I don't quite understand the difference between the PostgreSQL concepts of a cluster and a server, but it seems to me that they are the same thing?
tl;dr

Yes.

Confusion arises from the official terms used in the SQL standard versus more commonly used terminology. 


Cluster = database server
Catalog = a database
Schema = a namespace, a grouping of tables


Computer > Cluster(s) > Database(s)/Catalog(s) > Schema(s) > Table(s)

See my Answer to a similar question on Stack Overflow for more info.

Briefly here…

An installation of Postgres on a computer is called a cluster, per Section 4.13 of SQL-92 titled "Clusters of catalogs".

That cluster has one or more databases, each database called a catalog per the SQL standard. Each database contains one or more schemas, each schema being a namespace and security boundary for a collection of tables.

You can have more than one cluster on a computer. This is typically done for different versions of Postgres. For example, you could have Postgres versions 9.5, 9.6, and 10 all installed, each listening on a different port for incoming connections. That would be three clusters in that computer, each containing catalogs that contain schema that contain tables that contain your rows of data defined by columns.