These terms are confusing, misused, and inconsistently defined.

For MySQL:

A Database has multiple definitions:

• A "schema" (as used by other vendors/standards). This is a collection of tables. There are one or more "databases in an instance.
• The instance. You should use "server" or "database server" to be clearer.
• The data. "Dataset" might be a better term.

An instance refers to a copy of mysqld running on some machine somewhere.

• You can have multiple instances on a single piece of hardware. (Rare)
• You can have multiple instances on a single piece of hardware, with the instances in different VMs or Dockers. (handy for testing)
• Usually "instance" refers to one server with one copy of MySQL on it. (Typical for larger-scale situations)

A PARTITION is a specific way to lay out a table (in a database).

• It is seen in CREATE TABLE (...) PARTITION BY ....
• It is a "horizontal" split of the data, often by date, but could be by some other 'column'.
• It have no direct impact on performance, making it rarely useful.

Sharding is not implemented in MySQL, but can be done on top of MySQL.

• It is also a "horizontal" split of the data, but in this case across multiple "instances".
• The use case is, for example, social media where there are millions of "users" that are mostly handled by themselves. That is, most of the queries focus on a single slice of the data, hence it is practical to a bunch of users on one server and do all those queries there.
• It can be called "horizontal partitioning" but should not be confused with PARTITIONs of a table.

Vertical partitioning is where some columns are pulled out of a table in put into a parallel table.

• Both tables would (normally) have the same PRIMARY KEY, thereby facilitating JOINs.
• Vertical partitioning would (normally) be done only in a single "instance".
• The purposes include splitting off big text/blog columns; splitting off optional columns (and use LEFT JOIN to get NULLs).
• Vertical partitioning was somewhat useful in MyISAM, but rarely useful in InnoDB, since that engine automatically does such.

Replication and Clustering

• Multiple instances contain the same data.
• Used for "High Availability" (HA).
• Used for scaling out reads.
• Orthogonally to partitioning or sharding.
• Does not make sense to have the instances on the same server (except for testing/experimenting/staging/etc).

Partitions, in terms of MySQL and PostgreSQL feature set, are physical segmentations of data. They exist within a single database instance, and are used to reduce the scope of data you're interacting with at a particular time, to cope with high data volume situations.

The document you're quoting from is speaking of a more abstract concept of a data partition at the system design level.