How to Write to a Custom Component Attribute

I spent a long time trying to figure out why the following code was not working:

<apex:component id="this" controller="MySiteInputAccountIdCtrl"
selfClosing="true">

<apex:attribute name="value" type="Id"
description="Account Id to pass back to the page."
assignTo="{!accountId}"
required="true"/>

<apex:inputText id="accountIdIText"
value="{!accountId}"/>

</apex:component>

accountId was a simple property with a vanilla pair of getter and setter methods. When the component is rendered as a component like <c:MySiteInputAccountId value="{!contact.AccountId}>, contact.AccountId would not update no matter what I typed into the input field.

Several hours later, I took a look back at a custom component that worked (and was also created by me after a similar bout of confusion)... And it turns out that I was writing the input to the wrong object. The correct component markup is as follows:

<apex:component id="this" controller="MySiteInputAccountIdCtrl"
selfClosing="true">

<apex:attribute name="value" type="Id"
description="Account Id to pass back to the page."
assignTo="{!accountId}"
required="true"/>

<apex:inputText id="accountIdIText"
value="{!value}"/>

</apex:component>

I can't believe I forgot this resolution to a 3-hour frustration almost immediately... just to experience a new 6-hour frustration on the same topic. Hopefully writing this down will help me remember my lesson and avoid a third incident.