Going through Metaprogramming Ruby, and in this book, they went with
eigenclasses. On page 119, Paolo Perrotta says

<<QUOTE
The name eigenclass has an eventful history. Each Ruby programmer seems to
have a pet name for these entities. Most people still call them singleton
classes, but this name confusingly recalls the (unrelated) Singleton design
pattern. Other people call them "metaclasses," meaning "the class of a
class." This is still a fashionable name for eigenclasses of classes, but it
doesn't really fit eigenclasses of objects.

Yukihiro "Matz" Matsumoto, the author of Ruby, hasn't announced an official
name yet -- but he seems to like the mathematically-friendly name
eigenclass. The German word eigen roughly means "one's own," so the common
translation of eigenclass is something like "an object's own class." This
book sticks with Matz's vocabulary, but be aware that the term eigenclass is
not as widely used as singleton class.

I also faced another terminology problem while writing this book: what do I
call the methods of an eigenclass? Neither eigenmethods nor eigenclass
methods is easy on the eye. After a lot of arguing and coffee drikning, I
decided to stick with Singleton Methods, which is still the most common name
for these things
QUOTE

Okay, so, a few questions, then.
1. Is this already outdated, now that there is Object#singleton_class ?
After this thread, I had taken to calling them singleton classes, but then I
got to this spot in the book, and now I'm not sure again.
2. Why do people say that singleton class is unrelated to the singleton
design pattern, when the singleton design pattern is a way to define a class
with only one instance, and an objects singleton class has the object as its
only instance?