There are this two names: a subshell and a child-shell.

Yes, a child process will be started by any of this:

sh -c 'echo "Hello"' ( echo "hello" ) echo "$(echo "hello") echo "hello" | cat 

Are all equivalent and share the same name? Do all share the same properties?

POSIX has this definition:

A shell execution environment consists of ....

But the last paragraph of above link has this:

A subshell environment shall be created as a duplicate of the shell environment, except that signal traps that are not being ignored shall be set to the default action.

And specially:

Command substitution, commands that are grouped with parentheses, and asynchronous lists shall be executed in a subshell environment. Additionally, each command of a multi-command pipeline is in a subshell environment; ....

The sh -c 'echo "Hello"' is not included there, should that be called a subshell also?

There are this two names: a subshell and a child-shell.

Yes, a child process will be started by any of this:

sh -c 'echo "Hello"' ( echo "hello" ) echo "$(echo "hello") echo "hello" | cat 

Are all equivalent and share the same name? Do all share the same properties?

POSIX has this definition:

A shell execution environment consists of ....

But the last paragraph of above link has this:

A subshell environment shall be created as a duplicate of the shell environment, except that signal traps that are not being ignored shall be set to the default action.

And specially:

Command substitution, commands that are grouped with parentheses, and asynchronous lists shall be executed in a subshell environment. Additionally, each command of a multi-command pipeline is in a subshell environment; ....

The sh -c 'echo "Hello"' is not included there, should that be called a subshell also?