At a Home Inspection, paid for by the buyer, of course the buyer is going to be there! And, if the agent isn't incapacitated, the buyers' agent should be there as well.
If problems are expected to be found, (i.e., in an older, home with lots of 'deferred maintenance') , when it is one of my teams listings, the listing agent is in attendance as well. The reason for this is to be absolutely clear on the deficiencies pointed out by the inspector. Their reports, while comprehensive, are often hard to read; and their literary skills are often lacking as well. Many do not annotate their reports with photos.
The one person who should NOT be there, as a usual thing, is the seller. <Unless he is paying for a pre-contract inspection.> Why? Because any statements that the seller makes to the buyer, particularly if witnessed, might be taken as enforceable.
For example, if the seller were there in the above example (small cracks in basement wall) and the owner said he would take care of it. The owner meant he would paint over them to seal them; whereas the buyer might think he meant rebuilding the basement.
SO, to answer your question, in my opinion, at an inspection; the Buyer, the Buyer;s Agent, and the Listing Agent. The Seller should not be there, but available via phone to answer any specific questions the Listing agent might call with.
However, for this second, structural inspection, I think that you, since you are paying for it, should be there. I am less sure that I would have the buyer there, then the previous letters. NOT that I recommend hiding anything, but, if the cracks turn out to be of serious import, you will have greater worries than this particular contract. It would give you time to consult with your agent as to the best strategy to present the findings in a way to reassure the buyer.; normally, you present the problem AND the solution, at the same time.