I am going to make a bit of an odd suggestion. But first, I'll explain my criteria for buying it. I hate getting right up to the fence to video an at-bat, particularly at a place like Lake Point or at a showcase (I prefer to be invisible). My phone camera just doesn't have enough zoom to get this job done. Second, my phone was getting filled with video so I wanted something that stored the data off my phone. Third, I use my phone during a game so this gave me another reason to get a separate camera.
I ended up getting the DJI Osmo, which is a drone camera on a stick. I met a filmographer who was using one and he raved about it. Fantastic zoom (72x), takes HD video (60 frames per second), stores video on mini SD card, and has an amazing stabilizer that works great for baseball. It is really professional quality video that makes great clips for a recruiting video. Also, your phone is the "view finder", so you can set up the camera on a tripod and control the action from your seat in the bleachers. It's not cheap, but fit perfectly into what I was looking for.