The problem with the primary key violation looks to be where the provider is running:

update {databaseOwner}{objectQualifier}ModuleSettings
set    SettingName = 'HtmlText_ReplaceTokens'
where  SettingName = 'TEXTHTML_ReplaceTokens'
GO

Turns out that there may be instances where the SettingName is already HtmlText_ReplaceTokens but it also has a duplicate entry for the same module using TEXTHTML_ReplaceTokens.  I did a simple search in my database

select count(moduleid), moduleid from ModuleSettings where settingname = 'TEXTHTML_ReplaceTokens' OR settingname = 'HtmlText_ReplaceTokens' group by moduleid order by count(moduleid) desc

 And found a whole bunch with 2 entries (one for each).  I went and did a delete of all the ones with the old entry and then reinstalled the HTML module v5.1.3 and voila!  Looks like a past install (5.1.1, 5.1.2?) dorked up the settings and didn't do anything to get rid of the old settings.

The setting simply holds the state of the "Replace Tokens" module setting, which defaults to false. In most cases there is not change, if you delete the setting in the database, if you have instances, where you are using this feature, you should review module settings after performing an app restart.