First thing... The Mishnah Berurah (including the Be'ur Halakhah) has a number of idioms for soft stringency, ie for presenting a stringency as a desired objective but not as baseline law. Some statistics from Benjamin Brown's article in Contemporary Jewry (table on pg 2 of the PDF):
- Yeish/tov/nachon/raui lehachmir: 458 times
- ... lachush: 223
- ... lizaheir / lehizaheir: 104
- ... limnoa' / lehimana: 36
- Yarei Shamayim yir'eh (or another verb) lehachmir: 35
and the oft-quoted baal nefesh yachmir being used the fewest (!):
Total is 885 soft stringencies. By Comparison, the Shulchan Arukh with Rama invoke these soft stringency terms a total of 104 times. The Mishnah Berurah's contemporary, the Arukh haShulchan (most of which predates the MB, but Arukh Chaim does not), uses them 274 times, in roughly the same proportions.
Oliver's answer addresses the soft stringency concept in general, and I would have posted something similar about Rav Amital, citing a sichah of his that was emailed out by Yeshivat Har Etzion's (Gush's) Virtual Beis Medrash.
So, I will just add something about "ba'al nefesh yachmir" in particular.
The first usage in the major codes is the Beis Yoseif. It's not in the Rif, Rambam or Tur. The Beis Yoseif was written by Rav Yosef Caro (who later wrote the Shulchan Arukh), who was among the Qabbalists of Tzefas. So when he says "nefesh", he means "soul" with the connotations particular to the word "nefesh" in Qabbalistic use. In contrast to other possible terms for soul, such as ruach or neshamah.
Nefesh connotes the soul in its basest, most animalistic form. Animals too have nefashos -- which is why the Torah explains (?) the prohibition against consuming meat with circulatory blood in it by saying "ki hadam hu hanefesh -- because the circulatory blood is [of?] the soul" of the animal whose flesh one is eating. (Devarim 12:23, c.f. Vayiqra 17:11 for different wording.) Soul as life-force, the impact of living in a body and the power to do that living.
A ba'al nefesh is not everyone who is spiritually inclined. That would likely be the yarei shamayim. Nor is it someone who is worried about legal concerns regarding rejected theory, as there are numerous other terms for that. It is someone trying to overcome their physicality, meaning their bodily desires and laziness.
And this is consistent with the Mishnah Berurah's usages. For example, the first usage: The Shulchan Arukh says that David did not to take a nap during the day that is than the time it takes to make 60 breaths (Orach Chaim 4:16). The Be'ur Halakhah says it's either 3 hours (based on narrative about the Ari's Shabbos nap), a little over 30 min, or 3 min. And then he says the ba'al nefesh should try to minimize according to his abilities.
Mishnah Berurah 299:40 (on OC 299:10) says that a ba'al nefesh should not be lighting a lamp immediately after Shabbos (with a "Barukh haMavdil) but should wait until after UVa leTzion. (Although the gabbai of the shul should not be stringent in this way, so that he can light the lamps before Borkhu so that people can see their siddurim.) For the ba'al nefesh, wanting more light isn't reason to skimp on holding onto Shabbos.
So it would seem to me that "ba'al nefesh yachmir" means "someone who is trying to overcome / who has overcome their physical side, should be stringent" and not give it a foothold on this issue.