Speaking of premier elements 13
Please refer to my posts numbered 2 and 3 in the link that John has provided in his post (numbered 2 here). My posts in his link give full details of the Adobe watermark and the ways that it can be removed only from the tryout project file only after purchase of the product. This matter is associated with Premiere Elements versions, not just version 13.
And, I point out, the Adobe watermark cannot be removed from exports from the tryout project file.
If you have any questions or need clarification on what I have written, please do not hesitate to ask.
I had the same issue and one of these posts solved it but I don remember who.
After buying the license, open the project started with the trial version.
Go to TIMELINE>DELETE RENDERED FILES and select it.
SAVE and then CLOSE the project. These 2 steps may not be necessary but I did them anyway
Re OPEN the project and the banner will be gone.. Re-render the project and away you go