Answering my own question after thorough research:
First, I noticed that there is no consensus on where the UV spectrum ends and the blue portion of the visible light spectrum begins.
According to the WHO, as well as to most other sources, the UV spectrum covers the wavelength range 100–400nm (1). However, a few other sources, notably the SCENIHR, define visible light as starting at 380nm (2), hence UV light would be 100–380nm.
Another source, the ICNIRP, directly mentions that the boundary is not clear-cut, "the band 315 to 380–400 nm is designated as UVA" (3) . At least one source refers to the 385–405nm band as the "UV/Visible Radiation Boundary Region" (4).
To address the part about the dark tint of lenses causing pupil dilation, it is first worth noting that not all lenses that filter out UV light have a dark tint.
In fact, clear lenses can actually offer up to UV400 protection (5). There is even some research specifically on UV protection by clear lenses (6).
Back reflection of UV light
UV380 and UV400 are not the only eyeglass specifications relevant to UV protection. There is also eye-sun protection factor (E-SPF®), which was introduced in order to also account for back reflection of UV light from eyeglass lenses to the eyes (7). Therefore, this back reflection should also be taken into account when examining the effects of 380–400nm light exposure on the eye.
The 380–400nm Band
Now let's get to answering the question.
According to the Texas Technical University Health Sciences Center, 40% of the radiation harmful to our eyes falls within 380–400nm, which they call the "UV protection gap" (8). Another source states that, while 100–380nm light is "very harmful", 400–460nm blue light can still cause retinal damage and macular degeneration (9). The American Optometric Association also states that blue light, which they define as the range 380–500nm, may cause eye damage (10).
An article by ZEISS Vision Care explains that the UV380 standard was a "pragmatic choice" by the ophthalmic industry, and does not reflect "biomedical realities" (11). The article refers to the 380–400nm spectrum as the "most intense portion of solar UVR on Earth". I assume the author is referring to the physical definition of intensity of radiant energy, which is the power transferred per unit area, however I am not 100% sure that this is the intended meaning of "most intense".
I also found an article claiming that the WHO recommends UV protection up to 400nm (12), which again implies that 380–400nm does cause significant retinal damage. However, I couldn't find the actual WHO publication stating the recommendation.