After playing with the data for a bit, I found one set of data that has a downward trend. The RSS MSU lower trop global mean. Starting in 1999, which is a more reasonable year, it shift to a slight upward trend. Unfortunately your site doesn't give a value for the slope. Eyeballing it, 2002 or thereabouts is a better start, which drops them below the 15 year cutoff, except the line from NOAA didn't exclude peak to trough periods.
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.