Luqman in the Quran is described as a wise man and in these verses or the story which the Quran tells us about him he is teaching his son (and us = the reader) a lot of things about believe and the etiquette (Aadaab آداب) on how he -as a believer- should deal with people, how he should act, walk and speak etc. ...
Be moderate in your pace اقصد في مشيك: has been mostly interpreted literally in the tafsir books by the explanation: "don't walk neither fast nor slow".
Some scholars like Sheikh Taher Ben Achour added that moderate means moderate between a walk of pomp التبختر (an arrogant walk, who walks very slowly -to be seen-) or creep الدبيب (a shay walk, who walks very fast- to not be seen-).
In tafsir al-Bahr al-Moheet there's a quote of ibn Masu'd (May Allah be pleased with him) telling us that they were asked not to walk at a slow leisurely pace like Jews nor to walk along with the body close to the floor (quietly or secretly) as the Christians, and take a way in between both.
وقال ابن مسعود : كانوا ينهون عن خبب اليهود ودبيب النصارى ، ولكن مشيا بين ذلك
At-Tabari instead said, that Luqman has prohibited or discouraged his son here from arrogance, but he also added an evidence for not walking fast. This would go ahead with the meaning of this sahih hadith.
Al-Baghawi added to the moderate way of walking a quote of 'Ata': walk with calmness and ease.
وقال عطاء : امش بالوقار والسكينة
He also referred to (25:63):
And the servants of the Most Merciful are those who walk upon the earth easily
From this we can conclude a believer should walk neither fast nor slow, neither in an arrogant style nor a shay way he should walk with ease.
See also this fatwa which includes a comment from tafsir al-Qurtoby on the Verse 31:19. And this summary about Luqman and his advices.
And Allah knows best!