After testing on my touchscreen with an input pen as well, which turned out even more chaotic (triggering three times every now and then...), I'm quite sure it's not a code-problem.
But I solved the problem with set_pickable:
print ("Tapped! Timer started!") # <-- THIS WAS PRINTED 2 TIMES
yield (get_node("SpriteButtonAnimationPlayer"), "animation_finished")
I already did this with all other animations, but for some reason I thought I wouldn't need to with the extremely short "small" (0.001 sec)... duration obviously doesn't matter here!
Thanks for your thoughts!