He expected you to counter offer and when you didn't, it made him believe the price was too low.

I asked my agent for her opinion on my counter offer and she said that couldn't say much and that she needed to remain impartial. 

The owner made a counteroffer and we accepted.

There was no counter offer or anything (makes sense since it was bank-owned).

I believe the thing that made me question things was due to the fact that when I gave my agent the number I wanted to use in the counter-offer, she tried to convince me to lower my counter...which I found annoying and it infuriated my wife.

A loan will usually be better for you than a note sale.But it also sounds like you really need the cash out, if that is the case and you can't carry the note, hit them with a cash price as a counter offer.

You may encounter three possible outcomes when submitting an offer: acceptance, a counteroffer, or rejection by the seller.

I am writing a counter offer to have this clause removed.

I also never understood people who will not make a counter-offer, no matter the amount of the offer they are countering. 

I had 4 offers of 100K below list in the first 24 hours; so then I did a multiple counter offer on advice of the broker, reducing the list price by $50K, and got 3 takers on that; the broker  then recommended this buyer.I was unaware that my agent is also the buyers agent until this point in the proceedings, (although this info was present in the offers that were presented; I just didn't read closely enough), and it has made me feel weird although he has been very helpful and seems nice (only phone conversations).