HarmonyOS is built upon open-source OpenHarmony (which seems to have many variants, not sure all of them are open), and OpenHarmony runs on Linux / HongMeng kernel (see OSDI '24 paper for the latter, code not open yet)
Huweii's Harmony NEXT came about shortly after OpenAtom formed, which is sort of the Linux Foundation of Harmony. NEXT definitely has a lot of new & rebuilt aspects of Harmony, seems unlikely to be compatible or really the same; it's a major shift with new kernels & all.
I think HarmonyOS != OpenHarmony because the later still retains compatibility with Android while the former does not. I could be wrong.
The OS compatible with Android apps is HarmonyOS (<5.0), which you can literally think of as Android + some (maybe much) Huawei stuff. The thing based on OpenHarmony is sometimes referred to as HarmonyOS Next / 5.0, and some of those having upgraded to HMOS 5.0 are always complaining its compatibility in Chinese communities.
Huawei always invents so many names so that people often confuse them.