alt Hacker NewsWhen distros mention i386 support they often actually refer to i586 or i686, yes.
True i386 support would mean compatible with the original Intel 386 processor from 1985. The 486 added a few additional instructions in 1989 but things really changed with the Pentium in 1993 - that gave us i586 which is the bare minimum for most modern software today. Much software can still run on regular Pentiums today if compiled for it, but SSE2 optimizations requires at least a Pentium 4 or Core CPUs instead.
I play with retro PCs often and found OpenBSD's i386 target stopped supporting real 386 CPUs after the 4.1 release, and dropped support for i486 somewhat recently in 6.8. It now requires at least a Pentium class CPU, i586, though the arch is still referred to as i386 likely because it's a common proxy for "32-bit".
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Yes, 386 and 486 stayed relevant throughout the 90s because the price tag for new shiny processors is always higher, and it was not uncommon for customers to favor more or faster RAM/Disk space/graphic card/sound card (that was a thing back then) over better-looking CPU benchmarks.
The Linux kernel also requires at least i486 now. AIUI that decision had to do with smoothing out multicore/SMP support - which is a bit silly because no real 80386 systems in common use are even SMP, let alone multicore. But anyway.
Debian 3.0 was the last version that ran on the 80386 processor proper, but even as the CPU requirements for the "i386" architecture moved up to the 486, then Pentium, then Pentium II, the name stuck around. Partly from inertia, partly to not break the entire existing mirror infrastructure.