Chinese and Japanese's version are the same (in this case). Just different translation.
Actually, neither "Heat starts to die down" nor "Heaven and Earth begin to Withdraw" is a good interpretation of 天地始肅.
"Heaven and Earth begin to Withdraw" is a very literal translation. In this case, 天地 = everything = everthing growing from the ground, so 天地始肅 means plants are no longer sprouting.
"Heat starts to die down" is not a translation. Just a convenient English paraphrase.