In textbooks people claim that conversion of ADP to ATP as the biosynthesis of ATP using ATP synthase. IMHO, it is NOT ATP synthesis! It is just a phosphorylation reaction carried out by ATP synthase. ADP is recycled to ATP by adding phosphate group.

How ATP is in fact biosynthesised?

Let us see How Cholesterol is biosynthesised? You start from Acetyl-CoA and go on build every carbon present in cholesterol to biosynthesise Cholesterol using intermediates such as isoprenyl pyrophosphate and dimethyl allyl pyrophosphate and squalene and Lanosterol. Because it is a "de novo"  synthesis you need to start from simplest "piece" of carbon source and go on build the whole cholesterol molecule. Because it is a "de novo" synthesis you should NEVER EVER  use "cholesterol" as a catalyst to make cholesterol itself. The reason is that "Where does that "cholesterol" come from cant be justified.
Now, let us look at biochemistry books of de novo ATP synthesis starting from simple amino acids and go on cyclize it to for the "base skeleton" required for ATP.(rather AMP or ADP). In every synthesis reported in standard text books, they USE ATP itself to carry out the cyclization. Here is where I find it funny. Because YOU CAN NOT USE ATP to make ATP- it is as simple as that. Because the question arises "Where the heck that "ATP" come from?" Nobody seems to care about this flaw in biosynthesis of ATP! I have not seen any de novo atp synthesis without using atp itself as a "catalyst"! And again making ATP from ADP is not synthesis. ATP synthase is just an "recycling machine" by adding "phosphate group" that's all!