We’re a forum of Airbnb hosts, unfortunately not customer service agents who can help with your issue. All we can do is confirm your gut feeling: it does seem like you got scammed.
The order of things they asked you to do certainly sounds like a scam. Legitimate hosts only take payment through Airbnb and it’s fairly simple: you see all the prices up front and pay through the site.
The part about the landlord providing invoice details sounds like other scams I’ve heard of. It’s usually at this point that they send you to a website that is NOT airbnb. The length of your contract also makes this very suspect. Anyone who is doing long term rentals would not use Airbnb: 1) landlords want a real contract with deposits and background checks 2) They wouldn’t want to give Airbnb 12-20% of their 2 years of rental revenue.
It sounds like you checked with Airbnb to confirm that the web site was legitimate. Was it an official Airbnb domain (ie. Airbnb.com (or whatever country you’re in like airbnb.ca, not redirected to another site?)) If they told you it was the real site you should continue to lean on them to reimburse you.
Be sure you’re calling the ACTUAL Airbnb support and not a phone number a scammer may have provided.
The phone numbers are available here: Airbnb resolution center and contact information with phone numbers for most countries.