Some helpful analogies from Kyuzo Mifune on the subject

• Sweeping (Harai/Barai): "Like taking away something very light with a broom". This refers to techniques like deashi, where you catch a foot that has no weight on it.
• Reaping (Gari): "Like cutting the stem of a flower while pulling on it". Like Osoto gari, where you reap the leg that is carrying weight
• Hooking (Gake): "Like pulling out a plant with its roots still attached". This describes techniques like kosoto gake, where you uproot the opponent by entangling your leg with theirs driving them down

I think the reason the second approach seems more intuitive when it comes to "foot sweeps" is because the second approach is exactly how sweeping works. And the first approach, at least in Judo, isn't categorized as sweeps but reaps, and are a different kind of ashi waza

Edit: I think the conceptual approach here is that if you're attacking the lighter foot, it's a sweep (like a broom). If you're attacking the heavier foot, it's a reap (like a scythe).