Hi @bbenny7 : I'd consider a mixed model. The Matched pairs analysis (paired t-test) you carried out is a special case of this mixed model.
In the Fit Model platform, Value is Y, and the effects are Equipment and Sample. Sample should be considered a random effect (via Attribute menu) so that Sample to sample variation is removed and comparisons are made within samples.
And you may want to ask yourself if the difference (statistically significant or not) is meaningful. i.e., the confidence interval may be very tight, and it may not include zero (indicating statistical significant different)...but is that confidence interval a practically meaningful difference? For example, in your matched pairs analysis, the conf int on the difference is -2.41 to -1.15; that is a very small difference. From a practical/scientific point of view, is that difference (somewhere between -2.41 and -1.15) relevant?
Edit: Also, looking at the plots, it appears the data are rounded to 1 dp; it is not advised to analyze rounded data. The rounded data may be perhaps what you report, but for data analysis always use the number of decimal places to the precision of the measurement.