Does the Berne Convention automatically give copyright protection for new works that lack a copyright notice? For example, if I wrote and released a PDF book but forgot to include a copyright notice, do I still own the copyright to the book?
Copyright notice is not relevant to having a Copyright. 1
Copyright starts to exist the moment a work is created. When the pen touches the paper the first time or the hammer strikes the block to become a statue, the work is started to be created. At that moment 2, copyright is gained as it becomes a work, usually defined in the national laws. Publication for sure grants all the rights according to the Berne Convention rules - which are the absolute minimum standards.
Copyright protection subsists, in accordance with this title, in original works of authorship fixed in any tangible medium of expression, now known or later developed, from which they can be perceived, reproduced, or otherwise communicated, either directly or with the aid of a machine or device.
Similarly, the German Urheberrechtsgesetz does list what can be copyrighted and that the copyright is with the author, and presumes the act of creation creates the copyright for nationals, and publication for non-nationals:
Section 1: The authors of works in the literary, scientific and artistic domain enjoy protection for their works in accordance with this Act.
Section 2: (1) Protected works in the literary, scientific and artistic domain include, in particular: [List of items]
Section 120: (1) German nationals shall enjoy copyright protection with respect to all of their works, whether or not they have been published and regardless of the place of publication. In the case of a work created by joint authors (Article 8), it shall be sufficient if one of the joint authors is a German national.
Section 121: (1) Foreign nationals shall enjoy copyright protection with respect to their works published in the territory to which this Law applies, unless the work or a translation of the work has been published outside that territory more than 30 days prior to its publication within that territory. Subject to the same limitation, foreign nationals shall enjoy protection with respect to their works published in the territory to which this Law applies in translation only.
There is no registration needed to have a copyright.
However in the US you need to register a copyright to seek specific damages in the courts.
1 - Currently. In the past it had been very relevant, but laws have changed since then.
2 - Technically shortly after the moment that the first touch is done and the work gains some originality, but there already can be an artistic expression in a single brushstroke of a minimalistic piece or a single stroke calligraphy - and the absolute minimum originality needed is rather low: while putting ARD in a line in a specific font is not protectable under copyright (it's a trademark), a 6-tune jingle is enough to be deemed a protected work in itself! Under German law, this is the Kleine Münze
Yes, although it was not always so
The answer by Trish is quite correct, in current law copyright is automatic when any protectable work is "fixed in tangible form" (which includes a computer file). This is true for all countries that adhere to the Berne Convention or to the TRIPS agreement (created by the World Trade Organization or WTO). Almost every country in the world adheres to one, or both, of these. All countries with significant publishing industries do.
In the US, a copyright notice was required until the effective date of the 1976 Copyright act (which was in 1978). Any work published in the US prior to that date without a notice lost copyright protection permanently.
Countries that adhered to the Universal Copyright Convention but not to the Berne Convention mostly required a copyright notice. But all such nations have now (2021) signed Berne, or TRIPS, or both.
A copyright notice still has some limited legal effect. In the US it largely defeats a claim of "innocent infringement" which reduces damages in a copyright infringement suit if the infringer proves to the satisfaction of the court that the infringer did not know, and had no reason to believe, that the work infringed was under copyright. Thus all or almost all commercial publishers include a copyright notice on all new publications, and many individual authors also do so.