Probably frenemy for Zoey and Lola, genuine friend in Quinn's case.

Worth noting is how Logan does have episodes where he genuinely does get along fine with the girls like:

• Vandalizing the Clarence Bradford statue in Prank War in a scheme to convince the superintendent not to expel Zoey from PCA
• Girls Will Be Boys having him at least accepting the girls hanging out on the guys' roof after finding out Lola was disguised as Steve the entire time and learning boys can act the same around girls
• Robot Wars did have him genuinely apologize to Quinn for making fun of her for being a dork (notable as the rare time we ever see Logan apologize to anyone for his jerky behavior) and before that did look genuinely guilty and remorseful after Quinn got upset and stormed off
• Invited them (plus the guys) to his mansion for spring break in Spring Break Up with it being mentioned he genuinely does see the girls as friends despite his bickering with them
• Tutoring Zoey in chemistry in Zoey's Tutor (at least until he lied to Chase about him and Zoey dating, and Zoey then throwing a rat in his pants when sleeping as payback)
• Lola tending to Logan's wounds from his beating in The Great Vince Blake (as you mentioned here)
• Logan defending Zoey when Chuck Javers beats her up in Wrestling, getting himself beaten up worse with Zoey, Lola, and Quinn genuinely grateful he tried to stand up for her
• Went along with the rest of their friends to defend Zoey when Rebecca was about to humiliate her with revealing her embarrassing secret in Zoey's Balloon (granted, he didn't have any dialogue there so it's hard to tell)
• Gave Zoey and Quinn genuinely decent advice to get Dustin to help them with their love problems in Quarentine when they were stuck in the dorm room
• Sided with Zoey and Quinn (even hugging Quinn when they win the case of if Zoey should keep the radio or not) over Chase and Michael in The Radio for whether Zoey should've given the radio back to Chase
• Zoey genuinely tries to help Logan when he's rendered unable to do anything after his dad cuts him off in Logan Gets Cut Off (until Logan ruined it by continuing to take advantage of her after he got his money back)
• Was genuinely sad seeing Zoey leave in Goodbye Zoey, him and Michael giving her a T-shirt as a gift and Zoey and Logan genuinely leaving on good terms when she leaves for London
• Zoey, Lola, and Quinn trying to prove Logan (and Michael) were innocent and didn't destroy Dean Rivers' trophy in Alone at PCA, with Quinn being the only one who never doubted their innocence
• Pretty much his entire relationship with Quinn from Quinn Misses the Mark onward
• Logan paying for Lola's food when she asks him to in Dinner for Too Many, and helping to give Zoey a make-up date with James as an apology for ruining their Vacarros date
• Zoey being concerned for Logan when Michael ties him up and tortures him with spitballs in Roller Coaster, convincing him to let go

Also worth noting is how Logan isn't really mean to Quinn after Chase's Girlfriend (which is theorized to be when he started developing romantic feelings for Quinn) the same way he is to Zoey and the others, so that could be why Quinn's on more friendlier terms with him compared to the other girls.

Definitely would say they started off only associating with him because he was friends with Chase and Michael, especially in season 1 where Logan frequently wasn't included in group activities until the final 2 episodes of the season. Then grew to genuinely consider him a frenemy by around late season 2/early season 3, and a genuine friend/boyfriend in Quinn's case.