There is no template that you can modify, the rendering is controlled by the underlying <asp:Login control that <portal:SiteLogin inherits from.
On my machine it renders like this:
<div id="ctl00_mainContent_pnlStandardLogin" class="floatpanel">
<table id="ctl00_mainContent_login1_LoginCtrl" class="logincontrol" cellspacing="0" cellpadding="0" border="0" style="border-collapse:collapse;">
<tr>
<td>
<div id="ctl00_mainContent_login1_LoginCtrl_pnlLContainer" onkeypress="javascript:return WebForm_FireDefaultButton(event, 'ctl00_mainContent_login1_LoginCtrl_Login')">
<div class="settingrow idrow">
<strong>
<label for="ctl00_mainContent_login1_LoginCtrl_UserName">Email</label>
</strong>
<br />
<input name="ctl00$mainContent$login1$LoginCtrl$UserName" type="text" maxlength="100" id="ctl00_mainContent_login1_LoginCtrl_UserName" class="normaltextbox signinbox" />
</div>
<div class="settingrow passwordrow">
<strong>
<label for="ctl00_mainContent_login1_LoginCtrl_Password">Password</label>
</strong>
<br />
<input name="ctl00$mainContent$login1$LoginCtrl$Password" type="password" id="ctl00_mainContent_login1_LoginCtrl_Password" class="normaltextbox passwordbox" />
</div>
<div class="settingrow rememberrow">
<input id="ctl00_mainContent_login1_LoginCtrl_RememberMe" type="checkbox" name="ctl00$mainContent$login1$LoginCtrl$RememberMe" /><label for="ctl00_mainContent_login1_LoginCtrl_RememberMe">Remember Me On This Computer</label>
</div>
<div class="settingrow buttonrow">
<span class="art-button-wrapper"><span class="l"> </span>
<span class="r"> </span>
<input type="submit" name="ctl00$mainContent$login1$LoginCtrl$Login" value="Sign In" onclick="javascript:WebForm_DoPostBackWithOptions(new WebForm_PostBackOptions("ctl00$mainContent$login1$LoginCtrl$Login", "", true, "", "", false, false))" id="ctl00_mainContent_login1_LoginCtrl_Login" class="art-button" />
</span>
<br />
</div>
<div class="settingrow registerrow">
<a id="ctl00_mainContent_login1_LoginCtrl_lnkPasswordRecovery" class="lnkpasswordrecovery" href="http://test/m2365msnet35/Secure/RecoverPassword.aspx">Recover Password</a>
<a id="ctl00_mainContent_login1_LoginCtrl_lnkRegisterExtraLink" class="lnkregister" href="http://test/m2365msnet35/Secure/Register.aspx">Register</a>
</div>
</div>
</td>
</tr>
</table><span id="ctl00_mainContent_login1_LoginCtrl_ctl02" class="txterror" style="visibility:hidden;">Sign In Failed. You did not provide a valid email address.</span>
</div>
and the outer table can be eliminated under .NET 4 using the technique I mentioned.
I have no explanation how it can render differently for you on your local machine.
and when I visit your site that you initially posted http://sbid-dev.co.uk/Secure/Login.aspx
and I view the source I see the same thing I posted not what you posted.