So far my instinct tells me eyeeatingfish is probably correct on his understand of trespassing. But I am not sure I believe him on the detaining and searching issues. When strangers come into my yard to steal stuff the police always ask me if I want to trespass them. But I wonder if I am legally allowed to detain them forcefully, or even search them forcefully?

A house is a little different to a public space, That would be Trespassing in the first degree,

§708-813 Criminal trespass in the first degree. (1) A person commits the offense of criminal trespass in the first degree if:

(a) That person knowingly enters or remains unlawfully:

(i) In a dwelling;  or

(ii) In or upon the premises of a hotel or apartment building;

The warning is not "supposed to be communicated in writing" It is required BY LAW to be in writing. This is how it would be played out in court (i doubt it would even get to court), Your honor i was not aware that i was trespassing as the area was open to the public and no written notification was  provided to me as per HRS  708-814.
I dont know where you get the signing part from, You are not required to sign any trespass notification, Just being present with it is enough. 

The fact that you are leaving the store when you are asked for your receipt means you cant be trespassed anyway, as to be trespassing you have to be on the property after being asked to leave.

Trespassing is the criminal act itself, To be trespassing you have to of refused a request to leave. If you walk into a bank with sunglasses on and they ask you to take them off that does not mean that they can trespass you. If they ask you to take them off and you say no, then they say can you leave then and you say no, then they can have you trespassed.

Yes I know the law requires it, I was not denying that. What I was getting at with the signature thing is that absent a signature, how do you verify someone received a written notice? All the store has to do is say that they gave the suspect a written trespass warning but he threw it away. That is why a signed one with a photograph is proffered.

Being in the process of leaving the store does not make you immune to being issued a trespass warning. A trespass warning can be issued at any time. When people are arrested for shoplifting or police are called for some type of disturbance it is common to issue a trespass warning. It matters not whether the person refused to leave. The trespass warning is simply given because the store owner/representative wants the individual to not return to the property.

The criminal violation of Criminal Trespassing in the 2nd degree (with regards to a trespass warning) occurs when the person returns within a year or refuses to leave after given the warning.

So far my instinct tells me eyeeatingfish is probably correct on his understand of trespassing. But I am not sure I believe him on the detaining and searching issues. When strangers come into my yard to steal stuff the police always ask me if I want to trespass them. But I wonder if I am legally allowed to detain them forcefully, or even search them forcefully?

There might be a bit of confusion, the search issue is not connected to the trespass issue.

What I was explaining about the searching of bags by store security is that the 4th amendment does not apply to private parties, it only restricts the government from searching your property. If a store security detains someone as a suspect in a shoplifting, they can and do search the bags of the suspect so they can recover the stolen items.

That is not to say that anyone can search your bags at any time though, they have to be able to justify your detention. If they stop you but it is later found out they didn't have good reason to stop you then they may have committed false imprisonment or face civil issues.

But lets say you found someone in your enclosed back yard and it is clear they are not there for any good reason. You can detain the person until police arrive. Now during that detention if you decide to go through their bag then there is no 4th amendment issue. It might be good to have an explanation handy to show why you went through the guys bag. (ex. make sure he didn't steal something from you)

Everything has to be justifiable though so you might want to refrain from detailing the mail man

So what constitutes a good reason to detain someone? Is "that guy didn't let us check his receipt at the doorway" good enough?

So what constitutes a good reason to detain someone? Is "that guy didn't let us check his receipt at the doorway" good enough?

No, they would have to see not come in with the item, see you pick up the item, not pay for the item. They have to be 100 percent sure that you have stolen the item and be able to prove that, if not you could have them charged with  unlawful imprisonment

I really hate receipt checkers as I just want to leave the store with my stuff.  But I understand the reasoning to help prevent shoplifting  Stores may have to sell 10 things to make up the loss of one stolen item.

For commercial property there are 2 requirements for trespassing as discussed prior.  The store issues a trespass notice to the person banning them from the premises, if the person stays on the property at that time or returns in the future, then the police can arrest them for the crime of trespassing.  The store can detain anyone, however the store can be sued if no crime was committed..

Stores need to train their receipt checkers better.  Some will say it's illegal to leave without getting their receipt checked, others are threatening and rude.  I walked out of a store with my backpack and passed the receipt checker since I didn't buy anything.  Outside the store a worker runs up to my and grabs me on the arm and yells at me for leaving the store and says I could be arrested for doing that.  All he had to do was explain stuff to me politely and I'd have no problem with it.