I'm just curious.
No!
It would be very difficult to have no sugar...fruit has natural sugar and Type 1 diabetics eat fruit as a regular part of the diet. So no, what you asked is not true. In Diabetes 1 and 2, the pancrease is not working properly to regulate that amount of insulin produced while eating. Even with Type 1, sometimes the pancrease does work a little, but not efficiently enough.
Some diabetics eat sugary foods often, or drink alcohol and just compensate with increased insulin shots.
Neither of them have to avoid sugar, that's just a myth. They both should choose healthy meals.
A type 1 diabetic will be insulin dependent and will have to take insulin for whatever carbohydrate they eat. A type 2 diabetic may still be making their own insulin so they may choose to spread out their carbohydrates over time with smaller more frequent meals and or lower glycemic index carbohydrates.
That's not true and the reverse is not true. Type 1s and Type 2s eat sugar all the time. Sugar isn't just in candy or cake; sugar is in fruits and vegetables. While I don't eat fruit, I eat vegetables every day, so I am eating sugar (natural sugars) every day.
While no diabetic should just consume ridiculous amounts of sugar every day, diabetes isn't about not eating sugar. We diabetics are much more interested in all carbohydrates, including sugar, and to a lesser degree protein. Carbohydrates turn to sugar in the blood quite quickly. Protein also turns to sugar, but the process takes longer.
Often a sugar splurge is much friendlier on my blood sugar than a carby splurge, like bread or rice. The sugar spikes me, but my pancreas eventually responds and brings me down within a few hours. Bread and rice spike me, but it takes longer, the spike is prolonged, and I come down slower. So, I'd rather have half a chocolate bar than a couple servings of rice most days. I don't do either, but if I had to pick my poison...
No