I am working as a java professional and sometimes I also double up as a Oracle8i database programmer which I enjoy doing. I have written code for creating tables, used referential integrity, and have taken part in normalization up to 3rd NF.
The question is What I should do more to qualify as a database designer?
To really qualify as a database programmer in Oracle, you need to master PL/SQL and SQL. I suggest picking up a good book, such as Murach's Oracle SQL or an Oracle Press book on Oracle PL/SQL. You need to learn to create Oracle stored procedures and user-defined functions. You will also want to learn to use Oracle Packages as a way to modularize your PL/SQL application just as you would a Java application. If you are doing any programming, then learn the Oracle API for .NET with ADO.NET and/or Java with JDBC.
I'd also suggest subscribing to some of the PL/SQL forums on the Oracle website. It will also be a great resource if you get stuck with something or you just want to learn the best practices.
You mentioned you are working with Oracle 8i. That's a really old version. I suggest learning 10g at a bare minimum. You can download the a free version of Oracle, called Oracle Express, onto your own computer and play with it just as you would an Oracle server. Once again, you can find that on the Oracle web site. I really recommend the Murach's book as a good starting point. It provides a lot of exercises to practice what you learned.