"There has been an uptrend that is not significant" is more properly interpretable as "there has been no warming" than anything else.
Actually, no. "There has been no warming" is a positive statement, one that would need its own significance test. "No significant trend" means "the data over the last 14 years, taken in isolation, cannot provide conclusive evidence for or against warming." Which is quite different.
Now if you look in the previous record, you see that 14 years is simply too short a range to reliably detect significant trends, even when they are really there (as verified by using longer timespans). That's what Jones says in the bits you conveniently left out.
If the record was such that 14 years trend could predictably detect trends, then the absence of a significant trend in the last 14 years would be evidence against GW. Since they can't, it isn't. OK?
Now if the last 14 years' data cannot speak conclusively for or against GW, we need to ask the second best question, namely relative likelihood: given the recent record, even though no hypothesis reaches significance level, which is more likely than the other - warming, or no warming ? The "nearly-significant uptrend" is a coded way of saying that, even over the last 14 years alone, warming is "more likely" than non-warming, in the sense that if there was no warming going on, there would only be about 1 chance in 9 of getting similar or more extreme results.
If we add in prior knowledge, the overall long-term data says that warming is going on. The last 14 years of data, alone, cannot prove it, but they support previous data, rather than contradicting it, as you seem to imply.
tl;dr: "no significant trend over last 14 years" doesn't mean "no warming", it means "14 data points is not enough to establish significance in trends for noisy timeseries" (duh!).