Those two photos do not look like a Rumford. Not at all.
To answer, a Rumford is a design intended to project more of the heat out into the room rather than up the chimney. The sides slope in toward the back, so the floor of the fireplace makes a trapezoid. The rear wall, up maybe a third of the way, begins to slope forward. So, the sloping sides and back serve to reflect and direct heat out from the fireplace. They work great. I built ours by the formula 30 yr. ago and it will really heat up the room. The design came from a Count Rutherford in England [or was it Scotland?] who, or so the story goes, was concerned with the extremely smoky and ineficient fireplaces that would not really heat rooms even when roaring and wasting wood. There is quite an interesting story to all this if anyone cares to search it. The basic fireplace idea is to build them shallower in depth, taller, and with the aforementioned sloping sides and back. They are quite successful in this.
To answer, a Rumford is a design intended to project more of the heat out into the room rather than up the chimney. The sides slope in toward the back, so the floor of the fireplace makes a trapezoid. The rear wall, up maybe a third of the way, begins to slope forward. So, the sloping sides and back serve to reflect and direct heat out from the fireplace. They work great. I built ours by the formula 30 yr. ago and it will really heat up the room. The design came from a Count Rutherford in England [or was it Scotland?] who, or so the story goes, was concerned with the extremely smoky and ineficient fireplaces that would not really heat rooms even when roaring and wasting wood. There is quite an interesting story to all this if anyone cares to search it. The basic fireplace idea is to build them shallower in depth, taller, and with the aforementioned sloping sides and back. They are quite successful in this.