Is there any other way to use the Haskell type operator other than the type synonym?

The type and data that appear during class and instance declarations are GHC extensions called Type Families that provide a slightly different function than the type that appears at the top.
To enable the type family, specify TypeFamilies as the pragma (declared at the beginning of the linked source code):

Cf.https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/type-family.html

The class and instance declarations correspond to the usual class.

Normally, the class declaration will give you a method, its type, and default implementation.If you use the type family, you will be able to declare the "type-level version"That is the declaration of the type function using type.
In other words, you can define a type function, kind, and default type.

Here is a trial example to verify:

class Hoge i where
    type Fugai::* -- Species (kind) specified
    type Fugai=() -- Default type

    using piyo::Fugai->Fugai -- `Fugai`
    piyo=id -- Default Implementation

instance Hoge Inthere
    type Fuga Int=Maybe Int
    piyon=fmap succmn

instance Hoge Double where
    type Fuga Double = [Double ]
    piyols=ls++ls

piyo is the method and Fuga is the type function.You'll find that it can be declared in a similar way.You can also verify that Fugai is used as the type.

Let's look at the class declaration first:

class Storable (ImagePixeli) =>MaskedImage i where
    type ImagePixeli

This declaration shows that

• Defining class MaskedImage
• i is a type variable representing an instance of MaskedImage
• ImagePixeli must be an instance of Storable
• ImagePixeli is a synonym defined in the instance declaration
• kind is omitted
• No default type defined

That's about it.
Now look at the instance declaration:

instanceStorable p=>MaskedImage (Manifest p)where
    type ImagePixel(Manifest p) =p

This instance declaration shows that

• Manifestp is an instance of MaskedImage (the instance also happens to contain the type variable p)
• ImagePixel(Manifestp) is a synonym of type p
• p must be an instance of Storable

This instance declaration is valid if the type check passes.
However, Storable p may need a little more explanation.

The following is a brief confirmation that this instance definition is valid.

To make Manifestp an instance of MaskedImage,

• Rewrite class to instance
• i is mechanically replaced with Manifestp

provides:

instanceStorable(ImagePixel(Manifestp))=>MaskedImage(Manifestp)where
    Type ImagePixel (Manifest p)

The developer defined ImagePixel(Manifest p) as a synonym of type p:

instanceStorable(ImagePixel(Manifestp))=>MaskedImage(Manifestp)where
    type ImagePixel(Manifest p) =p

As defined by this type of synonym, ImagePixel(Manifest p) is replaced by p, so you get:

instanceStorable p=>MaskedImage (Manifest p)where
    type ImagePixel(Manifest p) =p