Seems to be I was a little confused
So here the correct answer:
Insert this line to xoops/html/include/registerform.php in line 77:
$reg_form->addElement(new XoopsFormText(_US_REALNAME, "name", 26, 25, $name), true);
the code in this area should look like this then:
$reg_form = new XoopsThemeForm(_US_USERREG, "userinfo", "register.php");
$reg_form->addElement(new XoopsFormText(_US_NICKNAME, "uname", 26, 25, $uname), true);
$reg_form->addElement(new XoopsFormText(_US_REALNAME, "name", 26, 25, $name), true);
$reg_form->addElement($email_tray);
....
You also have to insert this line into xoops/html/register.php on line 151:
<input type='hidden' name='name' value='".$myts->makeTboxData4PreviewInForm($name)."' />
So the code looks like this then:
....
<input type='hidden' name='uname' value='".$myts->makeTboxData4PreviewInForm($uname)."' />
<input type='hidden' name='name' value='".$myts->makeTboxData4PreviewInForm($name)."' />
<input type='hidden' name='email' value='".$myts->makeTboxData4PreviewInForm($email)."' />";
....
I don't know for which reasons the Real Name was left out from registration form, but I tried it on my local site and it worked.
Maybe there is someone who can explain?