I know that this sort of goes against the principles of a relational database but let me describe the situation.
I have a page where the user will place a number of
Use the Approach 1 and live with the performance implications of index updates. Unless you are dealing with millions of items per page, you are unlikely to find the performance lacking, and you retain all the power of SQL in dealing with sets of data.
In addition to being much harder to work with from the pure non-procedural SQL, the Approach 2 would still require you to traverse the list to find the right place to reconnect the "links" when reordering the item.