Magento – Display category products on homepage using carousel slider

This could be the problem:

<img src="../images/sale-icon.png" width="101" height="58" class="onsaleicon" />

When you are not on homepage the relative path to the image is wrong.
Use this instead:

<img src="<?php echo $this->getSkinUrl('images/sale-icon.png')?>" width="101" height="58" class="onsaleicon" />

and place your image in skin/frontend/{interface}/{theme}/images/.

Small tip:
Don't use this:

$specialprice = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialPrice(); 
    // Get the Special Price FROM date
    $specialPriceFromDate = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialFromDate();
    // Get the Special Price TO date
    $specialPriceToDate = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialToDate();

You are loading the product 3 times and you don't need to do that. You can load it only once if neccesary:

$mainProduct = Mage::getModel('catalog/product')->load($_product->getId());
$specialprice = $mainProduct->getSpecialPrice(); 
// Get the Special Price FROM date
$specialPriceFromDate = $mainProduct->getSpecialFromDate();
// Get the Special Price TO date
$specialPriceToDate = $mainProduct->getSpecialToDate();

but you can go without loading it, by editing the attributes 'special_price', 'special_from_date' andspecial_to_date` and settingused in product listingtoyes`.

bigger tip:

You can check if the product has a discount without checking the special price and the dates. This works even if you have a discount from a catalog rule:

if ($_product->getPrice() > $_product->getFinalPrice())

The three websites where it works probably have a bigger problem than the one where it doesn't: they do not have the recent security patch SUPEE-6788 applied.

One thing that the patch changed was that blocks to be used in CMS directives like that have to be added to a whitelist. You find this whitelist in System > Permissions > Blocks.

More details in this question: APPSEC-1057 How to add variables or blocks to the white list tables