Often, inbound emails include two parts: a plain text message, and an HTML message. The part we show by default is based on the plain text portion of the email. When you view the HTML for an inbound message, it should show the original HTML message, if one was included in the original email.
As for outbound messages, SmarterTrack handles those differently from inbound messages. Outbound messages are displayed as full HTML messages. This is done since we have more control over the safety of outbound messages than inbound messages.
In order to try and determine why the HTML view isn't working for the first example you provided, I will be sending you a private message shortly to request additional information.
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