It's a cone
In two out of three points of the spell description, the term used is cone.
The spell header has the following data.
Burning Hands; 1st-level evocation; Casting Time: 1 action; Range: Self
(15-foot cone); Components: V, S; Duration: Instantaneous
As you note from the text, it mentions both "cone" and "thin sheet" in the full description.
Two votes for cone, one vote for sheet.
Two or three dimensions?
In the spell geometry discussion we find this under cone:
A cone extends in a direction you choose from its point of origin. A cone’s width at a given point along its length is equal to that point’s distance from the point of origin. A cone’s area of effect specifies its maximum length. A cone’s point of origin is not included in the cone’s area of effect, unless you decide otherwise (Basic Rules, p. 80)
While that rules text may look like a two-dimensional shape on a grid, the illustration supports cone. (Three votes for cone). You could argue that it is not a three dimensional cone, but more like a truncated pyramid or a triangle when played on a grid. For theater of the mind play? It's a cone.
Plain English? It's a cone. Is the cone's whole volume necessarily full of flame? Unclear. See @Yakks' answer for why that may not be necessary.
DM call is needed here since the text isn't crystal clear
The rules text calls it a cone twice, a thin sheet once. A DM can rule reasonably that the three dimensional cone applies for a given situation like the one in the question you linked to. For most situations, the two dimensional and three dimensional cases are the same.
Area or Volume of effect?
Furthering the confusion, the text refers to the cone, and "area" versus "volume" for what is burned, or what is subjected to the magical effect. That may be done for ease of understanding by players and DM's who are not pedants, and who see a spell like burning hands (or fireball or cone of cold, as being in the class of "area of effect" spells rather than "single target spells."
As @DavidCoffron pointed out in a comment:
"area" in this case refers to the colloquial definition synonymous
with region rather than the mathematical definition. The area of
effect can have volume