Both Chelsea and Liverpool played good football in the first match of the season. But what happens is that Chelsea and Liverpool knows how to play draw in every of their meeting. So both of them playing together would not be the right criteria to determine whether the teams have improved. I can attest that Liverpool has improved right from the ending time of last season.
But there is no evidence of improvement in Chelsea apart from the pre-season. So we will need to see Liverpool play another club and Chelsea play another club for we to know if they have really improved.
Chelsea and Liverpool have been playing draws for a long time, and their first meeting of the season confirmed it once more. Liverpool significantly improved at the end of last season, while Chelsea also improved in preseason games. However, based on their overall performance following that game, I believe Chelsea outperformed them. But, as you mentioned, we'll have to wait and see how they play against other teams because theirs always ends in a tie before we can determine who is more powerful and improved this season. Nevertheless, one match can't prove that; we must wait for them to play 3-5 matches before we can say anything about them.
Chelsea and Liverpool have changed a lot this season, with the entry and exit of players making the first game a bit needing adaptation in the league, and a draw in the first game is a fair result for both teams, maybe with the next few matches we will see that the two teams will be more solid and shows the true quality of this team, and I believe in the second meeting in the league there will be a winner and judging from the first game I see Liverpool will be slightly above Chelsea at the end of the season