Wow, we definitely have a mystery! Those coordinates do not take me to the staircase or the portal room. Check out these screen shots. On one of the screenshots, you can see the seed - so I did copy it correctly.
Any thoughts? I'm in Australia, but why in the world would that matter?
Here is the last photo showing the coordinates where you found the portal room.
Your world must be older than you remember. If it was created back in the 1.8 series then strongholds would not necessarily be in the same place (and there were fewer of them). Testing that seed in 1.8.9 yields the same large mineshaft complex, but no stronghold. However, that also means you have been in the that area previously to generate the chunks without the stronghold (new chunks follow current version generation).
Could you recheck those coordinates of -1261, 708? I dug straight down and did not find the spiral staircase. It also didn't seem to be on the island. And yes, I would love the coordinates of the portal room, if you could include the level, I would greatly appreciate it.
Curious... how far did you dig down? I dropped into the spiral staircase at -1261 38 708
Wow, that stronghold is quite the mess, but an interesting one!
The stronghold is mostly overwritten by a huge mineshaft complex. It makes for a convoluted maze, but it looks like it would be both fun and challenging to explore. The two complexes are so intertwined that there are tracks on the stronghold bricks in some areas, and there is even a loot-chest minecart in the portal room!
The spiral staircase, which is generally considered to be the entrance to the stronghold, is under an island at -1261 62 708 (dig down from there). If you'd like the coordinates of the portal room, just ask.
Is this a world that was originally generated in an older version? If map or structure generation changed between the original version and the version you are playing now, it could affect where the eyes go.
What version was the world created in?
What is the seed, and coordinates the eyes are leading you to?