The fact that you lift your finger off the fret and there is still enough energy in the string to ring the harmonic pretty much says it all: you haven't damped the vibration. It is still vibrating.
An harmonic rings with a light touch. If you press harder it dampens it. IME. So when you are playing staccato just release the string just off the fret, keeping pressure on the string. If you are moving to another fret or string with that finger, your plucking hand needs to take over.
I don't know what to say. A ringing string is a vibrating string and you just have to kill it. I realize you asked that no one tell you it is technique, but a ringing string is a....you get the idea. You seem to have a bass with that elusive sustain everybody chases.
'73-4 Ampeg V4B, Ampeg PF350,PF115HE, PF115LF, 115 Traynor, 1962 Fender Jazz, 75 Gibson G3, '10 Squier VM Jazz, Squier CV 60's P Sonic, Squier CV 50's P blonde, Epiphone Thunderbird IV