Which one is it?

Serial #77778380 purchased in 2013 new from a AD in Australia

I don't know about years, but the reference number should be inside the case-back, I assume.

Superluminova dial. Serial in 77's. It's a 3570.50.00.

yep, mine.

77xxxxx are mostly 2005 if you go by Omega serial number charts. But as you can see mine was purchased in 2013

Firstly which online tools? ILMS only covers up to the mid 1990s and I don’t know of any other which accurately covers Speedmasters (or any other Omega model) post that. The online serial tables listed in various sources are next to useless for some models and eras, Speedmasters have been in the 77-79m region for ~20 years, immediately prior to that they were at 48m. These serials don't increment regularly like they do for other Omega series. Buy a Seamaster or Connie today, it may have a 90m serial or even an A0 serial, buy a Speedmaster it will likely say high 78m or maybe low 79m.

Secondly, the OP is getting confused by case number as seen in the caseback, here 145.0022, and PIC, in this case 3570.50.00. As mentioned above, if you have a Hesalite crystal Moonwatch produced from around 1996 up until the screwed bracelet suitcase model in ~2013 (with either tritium or SL dial) it will have case number 145.0022 and be described as a 3570.50. You might raise an eyebrow at my mention of tritium dial but the first year of 3570s were thus equipped. AFAIK the current suitcase model still has a caseback with 145.0022 etched on the inside, though the PIC is now the rather less memorable 311.30.42.30.01.005. Think of it this way, the case number describes the watch head, the PIC the full retail package inc bracelet/strap. I am not certain when the extra zero crept into the case number (145.0022 vs earlier 145.022), maybe when the 3590.50 model was introduced in 1989, but it was a good while back.

That's the wonderful thing about this hobby, the rabbit hole has no end.

I took a look at the website above, I haven't seen that before. It has many of the holes and issues the tables have but at least is easy to use. It does seem moderately useful for non Speedmasters made before 1990, prior to then the serials were more regular, so I thank you for posting that. I checked it against a few known serials of my own and it wasn't too far off on the earlier stuff. For Speedmasters, pre 1995 use Ilovemyspeedmaster.com, after that you need to ask for owners to chime in as the range is so narrow. A 77m could be 2000, a 78.8m could be 2018!