[JWait]

Umm.. are you trying to make "logout" one of your "tabs" because that is what it looks like?

By default "login" will display "logout" once a user is logged in.

Try changing..

{if $login ne ""}
<td>{include file="customer/tab.tpl" tab_title="<a href="include/login.php?mode=logout">$lng.lbl_log_out</a>"}</td>
{else}
<td valign="top"><a href="help.php?section=login" class="Tab">{$lng.lbl_log_in}</a></td>{/if}

to

<td class="Tab">{if $login eq "" }{ include file="auth.tpl" }
{else}{ include file="authbox.tpl" }{/if}
</td>

You will probably need to modify auth.tpl and authbox.tpl to suit your purposes, maybe even making a duplicate, calling them mod-auth.tpl and mod-authbox.tpl or whatever format you use for modified templates.

The reason you are not getting the desired response is that you are not getting the input to the proper form. You are calling include/login.php without including

<form action="{$xcart_web_dir}/include/login.php" method="post" name="loginform">

and

<input type="hidden" name="mode" value="logout" />
<input type="hidden" name="redirect" value="{$redirect}" />

Anyway, look at auth.tplo and authbox.tpl and you should find what you need.