I’'l try an explanation to the limits of my understanding.
Yes, the glut4 transporters on adipose cells are activated by insulin to allow glucose into the cells. The glucose is then converted into fat through a process called de novo lipogenesis (DNL) which is very similar to what your liver does, but produces a different fatty acid (palmitoleate). When insulin is high, glucose is stored this way and blocked from returning to the bloodstream. When insulin is low enough, the stored fat is allowed back into circulation.
After years of overstuffing a fat cell, and high insulin to keep it from being used, the cell eventually becomes enlarged (hypertrophic) and the glut4 transporter stops working. At this point, fatty acids flow back into the bloodstream and get converted to glucose again. This is when diabetes shows up as inability to control blood glucose levels after years of elevated insulin.
So, it may be possible that some of the gluose in the adipose cell did not go through DNL for whatever reason and remains there as a sugar. But I suspect a lot of the increase and fluctuations are caused by the outflow of fat as insulin decreases on keto or on a fast.