His personality was like this post-crisis, and even after Batman Year One. There was some variation of personality, and he had some more violent moments, but he spent a lot of stories like this, too. IMO. He was like this by two stories into Collins four-issue run.

Right after Millers Batman Year One ended in Batman 407, Collins wrote Jason stealing the tires origin in Batman 408. (Publishing date: June 87)

Meanwhile, as I understand it, over in Detective, #575 (October 87) is thought by some to be the first Post-Year One/Crisis deal Detective issue. (Depends on who you ask, but by some considered.) And above, you've got Tecs from 568-582. 575 was the first 'Tec after Barr's Year Two, so that's why some settle on that date as pseudo-official, but it wasn't a hard reboot for Bat titles, as I understand it was for Superman. Like you said, changes were different for Bat titles.

Some also mention the Legends issues, but at any rate, there was a lot of overlap.