N.B. IANAL (I am not a lawyer).
TLDR:
can [I] simply just add my COPYRIGHT WEBSITE NAME in my website
without any registration?
Yes, you can. You don't need to register a work to use the © symbol on it.
One reason you may be reading conflicting information is because:
- Copyright differs from one legal jurisdiction to the next.
See:
Copyright is a creation of law in each country, and therefore there is
no such thing as an international copyright law. Nevertheless, nearly
180 countries have ratified a treaty – the Berne Convention,
administered by the World Intellectual Property Organization (WIPO) –
that sets a minimum set of standards for the protection of the rights
of the creators of copyrighted works around the world.
Source: https://www.rightsdirect.com/international-copyright-basics/
How far your copyright extends and what actions you need to take to secure it will depend on where the work you intend to copyright is published.
For instance, in the UK, copyright operates as follows:
Copyright protects your work and stops others from using it without
your permission.
You get copyright protection automatically - you don’t have to apply
or pay a fee. There isn’t a register of copyright works in the UK.
You automatically get copyright protection when you create:
- original literary, dramatic, musical and artistic work, including illustration and photography
- original non-literary written work, such as software, web content and databases
- sound and music recordings
- film and television recordings
- broadcasts
- the layout of published editions of written, dramatic and musical works
You can mark your work with the copyright symbol (©), your name and
the year of creation. Whether you mark the work or not doesn’t affect
the level of protection you have.
Source: https://www.gov.uk/copyright
Further Reading: