I believe that neither Luffy nor Kizaru tried to kill each other in that fight. Let’s start with their first encounter, chapter 1091.

Kizaru’s Point Of View

Going into Egghead, Kizaru knew his mission, to defeat/ kill/ capture his old friends, which he now very clearly never wanted to do but had no say in the matter.

Another thing that Kizaru knew is that Luffy is protecting his friends.

Going into that clash, Kizaru most likely never had any intentions of taking Luffy seriously because he doesn’t want to complete his mission.

Luffy’s Point Of View

Luffy goes into that fight knowing that Kizaru is here to hurt his friends. Chapter 1091 ends with Luffy kicking Kizaru with Advanced Conqueror’s Haki.

It’s very clear that this wasn’t a serious fight from neither side. Kizaru’s focus is “securing Vegapunk” and now we know that he doesn’t actually want to hurt his friends.

Luffy’s Observation Haki

Luffy’s Observation Haki as explained to us by Rayleigh during his fight with Katakuri, tells us that Luffy’s Observation Haki is somewhat special in a way that allows Luffy to sense the feelings of others. It’s always been true throughout the series that Luffy is able to tell a persons true feelings despite how they seem to appear.

During the fight with Kizaru, I believe that Luffy understood Kizaru’s true feelings, maybe not in detail but Luffy’s haki may have allowed him to sense the hesitation in Kizaru’s actions and thus not taking him on as a true threat.

The conclusion to the fight

The fight concludes upon the arrival of Saturn.

When Saturn arrives, maybe Luffy understood the situation and that Saturn was actually the true boss and the reason for Kizaru’s hesitation.

At that moment he hits Kizaru with a star gun and gives Kizaru an excuse to get out of the fight. And Kizaru could also be the one who gave Luffy food because everyone on Luffy’s side was imprisoned by Saturn.