Please read through the instructions before trying them so you understand what needs to be done.
You likely did not install a program but ran a java applet which is how a lot of chats work.
I have never used this specific site but the chat would not be logged on your computer. Chats programs like skype and yahoo messenger do have the ability to save a log of chats to the computer itself however it is something you need to enable.
So no the chat you had on that site would not be saved on the computer however there could be evidence left behind on the computer that you did access that site and have a chat.
If this concerns you and you want to clean up any records of having used that site do the following.
Download the free version of Ccleaner here:https://www.piriform.com/CCLEANER
If you run the cleaner be aware it will wipe out all of your internet history, meaning we ALL websites visited and will erase all cookies so you will need log back into websites you were logged into, it will do this for all web browsers on the computer.
Now I would also suggest you clear your java cache as well.
The steps are a bit different depending on which operating system you use so follow the instructions on this page:https://www.java.com/en/download/help/plugin_cache.xml
You will see that you will be clearing java logs but don't worry, this is not an actual log of the chat it just logs the java program keeps.
After you have done both of the above if you have concern someone may snoop around and try and run some type of recovery tools then I would suggest you wipe your free hard disk space. This makes deletes files unrecoverable even with tools. That program Ccleaner I have you above can do this. Here is how you use that part of the program:https://www.piriform.com/docs/ccleaner/using-ccleaner/wiping-free-disk-space
But like I said, the chat itself would not be saved unless you specifically saved it. Unfortunately we cannot say if the person you chatted with copy and pasted the chat and saved it but I doubt it and since you used a codename it doesn't really matter anyway.
The steps I gave you above should cover you as far as any info on your commputer that you even went to that site.
In the future I would suggest you use a web browser like google chrome and take advantage of its incognito mode. In this mode nothing is saved to the computer once you close the browser.
Here is more info on that:https://support.google.com/chrome/answer/95464?hl=en
But just so you know almost all web browsers have a smilar private browsiong function where nothing is saved to the computer.
Please take a moment please rate my service by clicking on one of the smiley faces below. You can always return to this page to talk with me any time.