True, it’s actually just an ancient Chinese proverb (so not by Lao Tzu and can’t be traced to any single Chinese philosophers per se but most probably one of them during the Spring and Autumn period 770-210 BC, in which Chinese philosophy flourished).

The original quote is this:

“积行成习，积习成性，积性成命”

“Actions accumulate to become habits, habits accumulate to become character, characters accumulate to become destiny”

The current quote is apparently sourced to a Japanese scholar who studied ancient China (whom most likely was inspired by the ancient Chinese proverb); he’s been quoted to say something similar to this:

“心理变，态度亦变；态度变，行为亦变；行为变，习惯亦变；习惯变，人格亦变；人格变，命运亦变。”

Which directly translates to:

“Thoughts change, then attitude/mindset change; attitude/mindset change, then behaviors change; behaviors change, then habits change; habits change; habits change, then character changes, character changes, then destiny changes.”

Which I think then inspired the current English version