The first solution to this was to subtract a linear trend. However, global warming has not been linear over the last 130 years, so this unphysical procedure tended to mix effects of global warming and effects of the AMO. If you really want it, you can construct the index yourself by averaging SST over an area in the North Atlantic, and subtracting a regression against time.
Trenberth and Shea (2006) proposed to use the region EQ-60°N, 0°-80°W and subtract the global rise of SST 60°S-60°N to obtain a measure of the internal variability, arguing that the effect of external forcing on the North Atlantic should be similar to the effect on the other oceans.
Van Oldenborgh et al 2009 chose to leave out the tropical region, as this region is also influenced by ENSO. Guided by model experiments that show a low correlation between global mean temperature and variability in the overturning circulation (AMOC), they proposed to force the AMO index to be orthogonal to Tgobal by definition, lading to the second definition included here.
The indices differ on the monhly and interannua time scales, but agree very well on longer time scales.
If you would like to add another definition, please contact me.