The javax.faces.component.UIInput.CONVERSION message will be used when the ConverterException didn't contain a faces message and the component didn't specify the converterMessage attribute. In other words, it's the default/fallback message. The message identifier is specified by the UIInput#CONVERSION_MESSAGE_ID constant which is documented as follows:
CONVERSION_MESSAGE_ID
public static final java.lang.String CONVERSION_MESSAGE_ID
The message identifier of the FacesMessage to be created if a conversion error occurs, and neither the page author nor the ConverterException provides a message.
See Also:
Constant Field Values
The javax.faces.converter.* messages will be used when the in the message identifier specified standard JSF converter has failed a specific conversion task. The standard JSF converters are listed in class summary of the javax.faces.convert package summary. E.g. java.faces.converter.DateTimeConverter.* message identifiers will be used by DateTimeConverter which has some message identifier constants listed in its field summary like DateTimeConverter#DATE_ID with a constant value of "javax.faces.converter.DateTimeConverter.DATE" which is documented as follows:
DATE_ID
public static final java.lang.String DATE_ID
The message identifier of the FacesMessage to be created if the conversion to Date fails. The message format string for this message may optionally include the following placeholders:
{0} replaced by the unconverted value.
{1} replaced by an example value.
{2} replaced by a String whose value is the label of the input component that produced this message.
See Also:
Constant Field Values
Note that all message identifiers are listed in chapter 2.5.2.4 of JSF specification. See also JSF converter resource bundle messages for a copy.