I've never seen the spelling 'gantlet' in actual use, but apparently it's an American variant for both senses.
This maven wants to insist a gantlet is the thing you run, but I suspect someone's been browsing in dictionaries too much, and would like the differentiation because of the etymology.
These mavens more sensibly note just that there once was a preference for 'run the gantlet' among some American writers.
The Swedish
gatlopp 'lane-course' was assimilated to the glove as soon as it was borrowed: SOED gives 1646 for
gantlope and 1661 for
gauntlet.
The puzzling thing about this word is where the N came from, and one possibility is from 'gauntlet': that is there was very briefly an English word *'gatlope', a course to run, which immediately got influenced by the glove.
Another possibility is that the Swedish vowel caused it. It would have been [a] as in 'father', whereas the usual value of English A then was as in 'cat'. But before nC you often get or got AU: staunch, launch, daunt, and formerly such as advauntage; and while I'm not sure about the pronunciation of these in the mid 1600s, a representation of Swedish [a] with AU might have been normal, but AUT was not and it picked up an N from the general run of such words, rather than from gauntlet = glove specifically.
The spelling variations gantlet/gauntlet for both senses are just this spelling convention for the vowel: they're not particular to this word so they don't particularly depend on the etymology. There used to be lots more like advauntage. Many of them moved across to the CAT vowel [æ], and in 1800s Southern England many of them (aunt, plant, advantage but not ant, gantry, banter) adopted the [a:] vowel piecemeal. So modern pronunciations are often an unreliable guide to the etymologies.