Reddit mentions of Agloves Unisex Sport Touchscreen Gloves, Black, Small/Medium

Wool is good for Autumn, down-filled puffers are good for winter. I would reconsider your budget and maybe go for a fall ensemble first and save for a winter outfit. I think both of your coat choices are good, by the way, just not both for winter :)

Some things to consider:

• Getting a good pair of rain boots will not only help on rainy days, but you can also buy liners (like these) for wintertime. Rubber rain boots will hold up better than most other winter boots, especially when you start to get the slush and wet snow.
  
  
• As /u/peanutbutter_banana mentioned, the wind is killer here; invest in good, thick winter accessories. You'll need scarves, hats/earmuffs (this style wraps around the back of your head, which allows you to still wear a hat comfortably), and waterproof gloves. If you have a smartphone, get tech-friendly gloves so you can use the screen without having to freeze your fingers off.
  
  
• Socks: get some good quality wool socks. Wool will help wick moisture away and insulates better than cotton (this is actually an all-weather tip - thick wool for winter, thin for warmer weather)
  
  Source: I live right off the lake. I just moved back to Chicago after living in Arizona for 5 years, so I feel your pain!
  
  ^^^^Edit: ^^^^Added ^^^^link

They do make gloves that allow you to play PoGo while not unleashing your fingertip to frostbite

Bestdream Professional Joysticks for Mobile, Touch Screen Rocker Controller Mini Sucker Joypad for Smartphone Tablet Support Many Games https://www.amazon.com/dp/B074KDP948/ref=cm_sw_r_cp_apa_i_bPPTCbEZ1CGKX

Wired Gaming Controller, EasySMX PC Game Controller Joystick with Dual-Vibration Turbo and Trigger Buttons for Windows/Android/ PS3/ TV Box (Black) https://www.amazon.com/dp/B06XB6FRPR/ref=cm_sw_r_cp_apa_i_NPPTCb3AA75NM

Agloves Unisex Sport Touchscreen Gloves, Black, Small/Medium https://www.amazon.com/dp/B005G50S0Q/ref=cm_sw_r_cp_apa_i_jRPTCbBTBRGNJ