No problem. See this code from salutation.tpl:
PHP Code:
{if $salutation ne ""}
{$lng.eml_dear|substitute:"customer":$salutation}
{else}
{if $firstname eq ""}
{$lng.eml_dear_customer}
{else}
{$lng.eml_dear|substitute:"customer":"`$title` `$firstname` `$lastname`"}
{/if}
{/if}
In your example you are substituting "customer" with $customer.firstname.
When passed from /common_files/mail/html/order_customer.tpl the $customer array is not set. $order is what it is pulling from, as you can see from the scrap of code I posted above.
To accomplish your goal, I believe you only need to remove the reference to $title and $lastname in salutation.tpl. Like this:
PHP Code:
{strip}
{if $salutation ne ""}
{$lng.eml_dear|substitute:"customer":$salutation}
{else}
{if $firstname eq "" and $lastname eq ""}
{$lng.eml_dear_customer}
{else}
{$lng.eml_dear|substitute:"customer":"`$firstname`"}
{/if}
{/if}
{/strip},