I see I can write :
protected static
in my C# class (in my case, an aspx.cs). As well as :
private static
The definition of static isn't "available everywhere". It is a variable shared across the type it is declared within in the scope of an AppDomain.
Access Modifiers do not alter this definition, but obviously affect the scope of access.
You are confusing the static modifier with access modifiers. A static variable still needs accessibility defined. In your example, private static variables are only accessible within the type it is defined in, protected would be accessible within the type and any derived types.
Just a note, be aware that IIS (hosting ASP.NET applications) recycles worker processes, which will flush any static variable values that are alive at the time.