The issue that I am seeing in your thought processes is that you say delusion is a thought is believed and an obsessive or intrusive thought is something that is brought up that gives you anxiety. However, you need to understand that delusions are thoughts that are, like you said, believed by the person. Due to it being believed, the thought will rise to the surface during their daily life and will often be brought to their mind. And it's not because it is an intrusive thought but rather that they have often deemed unimportant observations to have meaning behind it. It's also important to remember that delusions don't have to give the person distress. A big example on this is delusions of grandeur or if we go back to endels, a person who simply believes they aren't human. It doesn't have to cause paranoia or bad reactions. They simply are unable to fully connect with reality.
Psychotic people with the help of outside influences such as therapists and loved ones and the usage of medication have the ability to recognize that certain thoughts are delusions but will still have these thoughts because, like I said, it's a chemical imbalance in the brain.
To go back to the neighbor analogy, the functioning psychotic is supposed to say ""I have a strong belief that this is true. However, I understand that the thought is highly unlikely and, therefore, most likely irrational. I am simply going to live my life with these beliefs in my head and not acting out on these beliefs." They will try to go on with their daily life.
Again, it's very similar to other mental illnesses in the sense that you can recognize thoughts are not good for your overall wellness but you will continue to have them. For example, someone with social anxiety will often have to process and try to ignore anxious thoughts about social situations. People with depression often have to deal with depressive thoughts but will still try to function as well.
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