You've made some mistakes with the Game of Life rules. You've missed out the overpopulation rule:

Any live cell with more than three live neighbours dies

Nit: > I guess there’s a harsh third rule which is, “if the cell is dead, it stays dead”.

That phrasing is inaccurate, if a dead cell stayed dead, the first rule wouldn't work. I'm not sure that particular sentence adds much to the flow, honestly.