In my opinion, the expression
1.5 times better
is to be avoided at all cost. It doesn't have any linguistic meaning. That is because "better" (like "bigger", "more") implies an addition to whatever thing is being measured:
this car goes 10 km/h faster than the other one
while "1.5 times" implies a multiplication.
(If we were to assign a meaning to it, because of the "additional" meaning, "1.5 times better" would mean "2.5 times as good").
Furthermore, if "1.5 times better" were to be allowed, what would "50% better" mean? When following the reasoning that would make "1.5 times better" mean "1.5 times as good", this would imply that it meant "50% as good". Which it of course never meant.
My suspicion is that the incorrect phrasing "1.5 times better" got into usage because people are unsure how to write "1.5 times as good as" and connecting it to the thing they are comparing.