I am a Realtor and have been in the business many, many years and closed hundreds of homes.
The reason you want a pre-approval letter to accurately reflect the amount you can afford (note, not the maximum amount, just an accurate amount) is that the seller will then do business with you in the case of multiple offers.
If you use the example above, if the list price is just under $140k and the buyer puts in his offer at $120k and his pre-approval states $120k - why would the seller even counter back? Lets say the comparable sales in the neighborhood within the last 90 days support a valuation of $130k and the buyer shows he can only pay $120k based on his pre-approval, even though the buyer can afford the actual value, the seller is very likely to reject the offer without a counter offer. If however the buyer has a pre-approval for $132k, for example, and he makes an offer at $120k the seller will at least pay attention to the offer rather than reject it.
The buyer is the one with the money. The buyer is the one that makes the initial offer. They buyer can reject any counter offer or walk away from any price that they are not willing to pay. So, why would you want a pre-approval to accurately reflect your ability? So your offer will be accepted.
One more reason to not play the game of putting in a pre-approval too low is that if you come back with a revised pre-approval from the same lender the seller gets a really bad taste in their mouth about your honesty. It looks like what it is: playing the seller for a fool. I've seen agents not work with the buyer in such a situation as the buyer is clearly not honest and therefore none of the information provided can be trusted.
As a buyer, you don't have to pay list price. You are the one with the money and the power to say no or yes to any counteroffer. The low pre-approval (one that is equal to the low ball offer price) is one way to make a deal go sour quickly.
BTW, the seller's incentive to accept your offer is to sell the house! They don't get the money without selling the house and then they can't move on. A seller is much more receptive to selling at a lower price to a strong buyer than a weak buyer. Usually a house is on the market because the seller wants to move to another house or another location.
Why would you show the seller what you are willing to spend on a house (or around an area near asking) and offer less for them to try and negotiate a higher price. I understand what you are saying but it makes no sense in letting them know that you are willing to buy for a higher amount or close to it. If they want to counter you that's fine. If they decline the offer why not have your agent negotiate a price? This is a business and its not dishonest in not letting them know how much you are actually willing to give for the house. By no means am I insulting your experience but I am questioning this technique of being honest to the seller. How are you being dishonest to them? The seller is trying to maximize money made and the buyer is trying to maximize savings. How does showing the seller how much you may be willing to spend then offering a lower price help you? It's a game of cat and mouse. I make an offer, they decline or counter, then my agent picks up the phone then negotiate a price in my favor. I never made an offer on an house showing I am willing to spend around 115k but will offer 100k and they listed at 120k. It's my money and if they want the sale they'll take it. Your scenario works best in multiple offer situations but if I'm the only offer on the table it doesn't seem like a good idea. It's like going to a yard sale offer less on an item and telling the seller you are willing to spend more but won't.