The problem with looking for information about stereochemistry on the internet is that many sources use outdated terminology, imprecise definitions or have downright erroneous information!
Here's a flow chart that I put together that I believe to be accurate, yet hopefully straightforward:
When using the flow chart, it is really important to remember that each decision on the flow chart is related to comparing two molecules, ie. it doesn't make sense to ask "what is compound A", but it does make sense to ask "how are compounds A and B related"? A molecule may be the enantiomer of another molecule, but a diastereomer of another.
Implicit in the flow chart is that stereoisomers have different 3-dimensional arrangements of their atoms, while having the same connectivity of atoms to each other.
The term "geometrical isomer" is now deprecated and its use is "strongly discouraged". It is an older term for cis-trans isomers, which are just a subset of diastereomers.
Back to your original post - the part in bold about diastereomers is problematic because the definition of diastereomers does not relate directly to stereogenic centers. It emerges that compounds with two or more chirality centers (the correct term for what are often referred to as stereogenic centers or stereocenters) can often exist as diastereomers, but chiral centers are not a pre-requisite. For example, cis- and trans-2-butene are diastereomers according to the definition.
A meso-compound is an achiral compound (the mirror image is super-posable), which is a member of a set of diastereomers, where at least one of those diastereomers is chiral. A meso-compound most commonly arises when the compound has two or more chiral centers, yet is not chiral due to an internal plane of symmetry. Meso-compounds don't fit neatly on the flow chart, which is based on comparing two molecules, but are a particular type of diastereomer.