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No, altitude should not affect the freezing point. Theoretically it can, but not at pressure changes we are likely to experience with different elevations. Contaminants (minerals or sugars) in the water will have an affect. For best accuracy, use distilled water and ice made from distilled water. The triple point for distilled water (when liquid, vapor, and solid states all occur simultaneously) is 0.01 deg.C
Most of the thermistors used on meat thermometers are not intended to be used for this kind of accuracy.