I think the more likely issue is just a matter of bias, not deliberate malfeasance. It's really hard to see errors in something when you get the results you expect. For TOBS adjustments for USHCN in particular, a mathematical artifact from the way they try to correct for it seemed more likely. That's based solely on my intuition as a mathematician though. Since we know they have engaged in deliberate efforts to conceal disconfirming evidence, and to punish journals for publishing papers whose conclusions they didn't like, I don't consider malfeasance completely implausible.
It's interesting to just plot the trendline of various sets. I especially find it interesting when you do so from the year I suggested previously, 2002.
CRU Global Average is how they described in on their page. Going by the link, it was HADCRUT3. Which shoes only
I meant to include this in my previous line. The way NOAA wrote that note in 2008 did not exclude peak to trough periods of 15 years or longer. It's a reasonable exclusion to make though, so I decline to hold that against them.
Being highly set dependent is suspicious, but the on the other side the adjusments made to surface station data are suspicious in nature based on my own checking. Specifically, averaging all adjustments for the GHCN after the flap about the Darwin station, I found a definite warming linear trend in the adjustments. The USHCN had a significant quadratic trend in adjustments, such that 80% or more of the warming trend in the data was created by the adjustments.
They were using global coverage. CRU Global average in the downward trend, and UAH in the one that had a very slight upward trend (.0004 per year). I can't see the slope of the trendline on that site you gave, but visually it's the same. Calling that basically flat is defensible, assuming reasonable error bars.
The simulations rule out (at the 95% level) zero trends for intervals of 15 yr or more, suggesting that an observed absence of warming of this duration is needed to create a discrepancy with the expected present-day warming rate.
Reading your link, their claim amounts to adding a new correction term to bring them in sync (that's what the new green line and range on the graph is). Which means they need another couple of decades to test the new modified models to see if they are sufficiently accurate. "We're right because we can add a correction term to cover this divergence that we didn't predict before" is entirely unconvincing. I find it highly suspicious that they chose to claim 2001 as the start, which doesn't match the date anyone else is using for the start of the current pause, 1999.
warming: increase in temperature
anomaly: difference in average temperature from a defined period (that's how the word is generally used in climate science)
A hiatus in growth of the anomaly really does mean a cessation of warming. It make pick up again later.