Personally I use the Terminal & SSH add-on from the official add-ons repository, the web based terminal saves you having a client so you can do stuff from your phone or tablet.
This is the one you have linked above.
But the best option is to take a snapshot before upgrade. Especially in your case where you have full access after upgrade. I have used the snapshot in the worst case scenario of a full rebuild after failed upgrade (well before they had safe-mode).
Hi, just did it and your code gave me:
The ‘hassio’ command is deprecated, please use the ‘ha’ instead!
So the one that has been working is ha core update --version=0.116.4
And if anybody is interested: in Hassio supervisor, I went to add-on store, then installed this addon “Terminal & SSH”. Started it and re-typed the code above.