Part 1 of the definition is more about agreeing to receive something, such as a gift or award. Part 2 means the same thing, but with ideas, such as accepting the truth of a statement or belief system.

It's entirely reasonable to be confused about this word. It confuses a lot of people; myself included. A lot of times in the context of injury or disability or tragedy a professional might ask someone whether he "accepts" the fact of his injury/disability/tragedy and it's not entirely clear what this entails. Obviously on one hand everyone recognizes the reality, but the term acceptance implies more than factual acknowledgment; it implies a certain outlook towards things which isn't entirely negative or nihilistic. What that outlook exactly is isn't entirely clear to me.

The topic is complicated.

:100:

Acceptance is the opposite of rejection, where to reject is to receive disfavorably ("Why did you give me this? I don't want this! Take it back!") or, of ideas, to disbelieve ("I reject your reality and substitute my own!")

Part 2 means the same thing, but with ideas, such as accepting the truth of a statement or belief system.

In that sense, what does it precisely mean to "agree"? because even though an idea seem reasonable and often even evidently feasible, there is struggle including that particular idea or rationale in one's belief system. So my question would be better formulated as: "What does it psychologically implies to accept something?"

Also, you would agree that although many people boast of having accepted an idea, they sooner or later come back to their initial belief system status, so to speak, thus, the fact of accepting the reasonable quality of something does not directly implies accepting it itself.

the term acceptance implies more than factual acknowledgment

Not exactly an acknowledgement but rather a recognition, acceptance implies more than factual recognition. I think acknowledgement would be far beyond recognition in the way of acceptance; as I think of it as a process, rather than an immediate ability.

Acceptance is the opposite of rejection

I really know about its opposite same as itself; only confusion.

Agree means sharing a certain belief with others. I would say that there are levels of acceptance. Consider the person who “accepts” a certain law, but nevertheless disagrees with it. On the surface this seems like a contradiction, but it’s not. The person is in agreement that the particular action is illegal, but disagrees that it should be illegal. The disagreement is over the fairness or justness of the law. So these types of examples could be one level of acceptance, whereas examples where there is no disagreement whatsoever would be a different level of acceptance.

Or, if you want to think about it in different terms, it’s possible to accept something cognitively, or intellectually, but not emotionally. I think the death of a loved one is a good example of this. You cognitively know that he or she is dead, and therefore accept that fact, but emotionally may be in denial.

1. Presented
2. Accepted (or Rejected)
3. Processed
4. Agreed/Disagreed (Belief/Disbelief)

So perhaps more along the lines of willingness to process when talking about a concept?

Consider the person who “accepts” a certain law, but nevertheless disagrees with it. — Pinprick

That depends on what one considers acceptance because, in that case, disagreeing is a direct opposite of believing, and in the most abstract sense, to accept is to believe.

note: not believe but to believe; and my confusion lies in what that distinction implies.

Or, if you want to think about it in different terms, it’s possible to accept something cognitively, or intellectually, but not emotionally. — Pinprick

I think what you mean by accept something cognitively and or intellectually is just recognition. I'm not sure what you precisely mean by emotionally... maybe to sublimate a suffering? because although I think of acceptance as a gradual process with several interrelated elements involved, it would be also not appropiate to infer its nature from the individual point of view of some of them.

So perhaps more along the lines of willingness to process when talking about a concept? — Regretomancer

I think this is a good starting point; I will try to figure it out on my own. However, do you think of it as something immediate?
and specifically in that regard, how would you define willingness?

Geez, I'm glad you all weren't around when I was growing up. Acceptance is one of the first things I learned. I learned early that hunger was gonna visit me pretty much every night. But I had no way of doing anything about it. I was 5' 11' / 3/4 and weighed 138 lbs when I graduated and put on 29lbs in
9 wks of Boot Camp. Acceptance has been a defining element in my wonderful long life. Brooding over what the problems of the world are is the greatest waste of the most valuable thing everyone has in their lives: Time. Hell, I'm 67 and I've got less Time left than all of you. I'm not going to waste my time fawning over the Worlds issues. I waited till my thirties to go to College and get a degree. Acceptance will calm the anxiety right of you. People will sense the fact that you aren't sweating over small things and people will find you personable, non-judgmental and pretty much agreeable to anyone.
I took one philosophy course in College and that was enough. If you want to learn understanding about existence and values and truth, the best method for learning all that stuff is to get out in it and enjoy it. You'll never feel the need to consider perspective again. You'll have the best perspective your Time can buy.

Live Long and Prosper.