'70 fuel gauge-weird problem

I figure you will wind up contacting the people that converted your gages to “digital”.
Or maybe someone has had this problem with theirs and can give a better idea than me.
This is a WAG. The only difference I can see in your description is voltage drop.
If you aren’t doing a full start load and the gage works.
And then you are going ahead and hitting the starter again and it continues to work, MAYBE the digital has some type of “memory” and you are tricking it by not starting the car on the first hit.
(Do I understand you correctly?)
I think “on” is “on” as far as the starter switch is concerned.
Don’t see any difference there.