The AMV index we reproduced in the blog is taken from the Zhang et al. (2019) review paper.  You'd have to provide a link to a source describing in more detail the "NOAAs data" so that the two series could be compared in more detail.  The image you show appears to have a smoother (lower-frequency) time smoothing applied.  Also different authors sometimes use different definitions to construct the index and different SST data sets.  In my initial comparison of the two images I don't see much of concern in terms of differences.  One series seems to be much smoother than the other (different filtering?)