I followed the article you listed: http://yoweb.co.uk/yo-blog/google-tag-manager-how-to-add-to-squarespace-template
It worked for me perfectly well. No issues on web, mobile, or tablet pages.
Here's what I did:
Step 1: Remember to change the 'XXXXXX' in the 'GTM-XXXXXX' to your GTM code though! For both Step 1 and Step 2
Paste this in the Header section of Code Injection in Squarespace.
Paste this snippet of code in the Footer section in Squarespace Code Injection.
Remove the Google Analytics Tracking ID if you've added it in the Settings > Advanced > External Services, Google Analytics area. This is not needed as the Google Tag Manager will directly be using the code you've just added to directly use Google Analytics data.
Voila! You're all set.
p.s Use Google Tag Assistant to check if your tags that you've set up in Google Tag Manager are working!
I referred to this video to set up a quick tag: