Some die hard Jeepers will blow a gasket if you say you're going jeeping. And I'm sarcastic enough to say it to them just to watch them get their knickers in a twist. But then they start in with the lecture about the use of the name and my eyes glaze over and I walk away.
Call it whatever you want. Call it "Jeeping". Call it "jeeping" with a lower case 'j'. Call it something else. Makes no difference to me.
If you want to go have fun driving over off road trails with me, you're more than welcome whatever you drive. If you get stuck, I'll drop a strap and tug you out. If I get stuck, I don't care what pulls me out. I love joking with people who have different brands than I do but we're all having fun and that's what counts.