Unfortunately, I can't really give you a clear answer. Maybe some of this information will help somebody else do so. It certainly seems that one of those websites is out of date. It's not entirely clear to me which. Various official sources have different information, I've emphasised in bold the conclusion that would be drawn from each of three official sources. They do not agree with each other, so please don't just read the first and assume that's the answer.
According to the "Visa regime for foreigners" page on the website of Albania's Ministry for Europe and Foreign Affairs, entry without a visa is available to several groups of people. One of those groups is:
d) Foreign citizens who have a valid permit of stay, issued by the competent authority of the United Arab Emirates
This seems to be your case. If this page is correct, your UAE residence permit is sufficient to enter Albania.
However, that part of the text refers to a law from (and was presumably last updated in) 2018. The page also links to this document, updated April 2021 (i.e. this document was updated more recently). This document indicates some groups of people are entitled to visa-free entry. UAE residence permit holders are not among them. If this (seemingly more recent) document is complete and correct, your UAE residence permit is not sufficient to enter Albania. (Family members of EU citizens are also listed on the previous page but omitted on this document - that strikes me as a less-likely change than the one relating to UAE residence permits, and may be a sign that the document is simply incomplete).
The website of the Albanian embassy in UAE notes that visa-free entry is available to holders of multiple-entry Schengen visas, but makes no mention of UAE residence permits allowing visa-free entry. I would imagine the embassy in UAE to be very aware of special rules for UAE residence permit holders (although it's worth noting that embassy websites, especially of smaller nations, are often poorly maintained). If the website of the Albanian embassy in UAE is complete and correct, your UAE residence permit is not sufficient to enter Albania.
In short: various official sources seem to disagree with each other. My first link is the one I would find more trustworthy than the others, but in the circumstances I wouldn't have enough confidence in it to simply assume it is correct. In your situation, I would contact the embassy directly to request clarification. I'd include the links above to make the source of confusion extremely clear.