The specific card that caused this question, is Milk of the Poppy, which explicitly says not to include Traits, doesn't say anything about Crests though and the manual doesn't specifiy either.

Hand of Osiris said:

Nope, crests are not part of the text box. Neither are challenge icons. From the FAQ, page 8:

(3.9) Icons in the Text Box
Gold modifiers, influence, and initiative
modifiers are not called out with an outline,
and are therefore a part of a card's text box.
Challenge icons and crests are differentiated
from the text box of a card by a solid outline,
and therefore are not considered a part of that
card's textbox.