Okay, after scratching my head for about 20 minutes, I think I've worked out what you want.

• If the first two characters are not underscores, but the third character is - then keep everything up to (but not including) the second underscore. Otherwise...
• Keep everything up to (but not including) the first underscore.

Is that correct?
If yes, then the following should do it:

for (@strings) {
    my $temp = "";
    if (($temp) = $_ =~ m/^([^_]{2}_[^_]+)_?/) {
        print $temp;
    }
    elsif (($temp) = $_ =~ m/^([^_]+)/) {
        print $temp;
    }
}
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Update: Oh, one thing I forgot to point out, was that there was an error in the code in my earlier reply. I hadn't accounted for the fact that there may not be any underscores at all. That's been fixed in the above snippet.