What is the major product in the reaction of 2,3-dichlorobutane with sodium amide in liquid ammonia?
These are both reasonable mechanisms, and the question outlines well the factors favoring each. In favor of mechanism I:


Low temperature suggests kinetic deprotonation
Statistically more terminal hydrogens than internal hydrogens


In favor of mechanism II:


Small base suggests thermodynamic deprotonation


In these cases where there are conflicting factors, I agree with @Lighthart that the best answer is to actually look at the experimental results. That will give an indication of how to weigh the various factors.

In this case, I do not find an exact hit in SciFinder, but the reaction of  2,3-dibromobutane does come up in two older reviews by Bergstrom and Franke. I do not have access to Franke's review, although SciFinder has a note that "Bromine represents any halogen."

The SciFinder result for the Bergstrom review is deceptive. There is a comment "generalized reaction, halogen and R-groups can vary," and I don't find the exact result (or reasonable analogs) in the review.

More broadly, there are seven results for an internal, vicinal (1,2) dibromide reacting with sodium amide to give an alkyne. One reliable example is the synthesis of stearolic acid from Organic Syntheses.

Similar SciFinder searches for the 1,3-diene product come up empty.

Based on this research, I believe the alkyne would be the predominant product.