You don't say whether you are using "one page" or "fast lane" checkout but I think what you are looking for is in the common_files/modules/ "whichever checkout you use"/ content.tpl. Its the part that says
Code:
<div class="checkout-buttons">
{if not $std_checkout_disabled}
{include file="customer/buttons/button.tpl" button_title=$lng.lbl_checkout style="div_button" href="cart.php?mode=checkout" additional_button_class="checkout-3-button"}
{/if}
{include file="customer/buttons/button.tpl" button_title=$lng.lbl_continue_shopping style="div_button" href=$stored_navigation_script additional_button_class="checkout-1-button"}
</div>
<div class="clearing"></div>
Note: the above code is from 4.4.3 but should be the same as 4.4.2
The other button is in skin/vivid_dreams_aquamarine/customer/main/cart.tpl.
Code:
<div class="right-buttons-row buttons-row">
{if not $std_checkout_disabled}
<div class="checkout-button">
{include file="customer/buttons/button.tpl" button_title=$lng.lbl_checkout href="cart.php?mode=checkout" additional_button_class="main-button"}
</div>
{/if}
{if $active_modules.Special_Offers}
<div class="button-separator"></div>
{include file="modules/Special_Offers/customer/cart_checkout_buttons.tpl" style="link"}
{/if}
</div>
<div class="clearing"></div>