Ok. here is what is happening:
The .htaccess file(s) provide directives to the Apache webserver that effect how the server operates on any directory it resides in and all directories beneath it. This can be over-ridden by another .htaccess in a lower level directory.
This is why your first domain works as you expect, but the second one didn't - the .htaccess file in your "main" directory (/home/username/) would impact the server operation for both the firstdomain.tld and the seconddomain.tld subdirectories except for the fact that it's operation was "over-ridden" for the firstdomain.tld subdirectory, because another .htaccess file (in that directory) "superseded" it.
As there was no such "superseding .htaccess" file in the seconddomain.tld subdirectory, whatever directives the "main" directory's .htaccess file contained were applied to the seconddomain.tld subdirectory.
That is why I asked about the contents of the .htaccess file - since you did not provide that in response to my question, I can not tell exactly what the directive was that caused the problem, but I suspect that it was an .htaccess file for another web application that prevented the index.html from being automatically displayed (several directives could be responsible for this).
All that said, and not knowing exactly what that .htaccess file in your "main" directory was designed to do or what it contains, I cannot advise you as to whether you need it or not - you certianly do not need it with it's current contents, if you want index.html to be automatically displayed when a browser visits seconddomain.com.
You can begin to learn about how .htaccess works by studying this Apache .htaccess "How To" (tutorial). Note that what you experienced is described in the section of the tutorial that states:
[i]"Apache must look for .htaccess files in all higher-level directories, in order to have a full complement of directives that it must apply. (See section on how directives are applied.) Thus, if a file is requested out of a directory /www/htdocs/example, Apache must look for the following files:
And so, for each file access out of that directory, there are 4 additional file-system accesses, even if none of those files are present. (Note that this would only be the case if .htaccess files were enabled for /, which is not usually the case.)"[/i]
I suspect that somehow the .htaccess file for your firstdomain got itself uploaded to the "main" directory, and caused the problem. I suggest you just leave it "renamed" in your "main" directory until you study the tutotial or, in some other way, determine what it is trying to do and how/where it should be placed if it is needed at all.
Leaving it where it is in a "renamed" condition will not hurt anything, and you may need to refer to it to figure out what it was trying to do.