I hate being wrong so I called in. I can confirm that Sears Club is different for Sears Card and Sears Mastercard. In fact, click here and it asks you how you will earn Sears Points before taking you to the redemption options. I had looked into this a year or so ago and I guess my recollection was correct.

So, the way I read it:

Sears Card Club points - 1 point per dollar spent at Sears and a few participating merchants (changing 1/Feb, see below). You can get the following rewards:

1K -> 20K Petro -> 2K CX
12K -> $360 Sears GC (3% return)

Sears MasterCard Club points - 1 point per dollar spent anywhere, 2 per dollar spent at Sears. You can get the following rewards:

1K -> $10 Sears GC (1% return on non-Sears purchases)

Now, Sears Card Club is changing to be more like Sears Mastercard Club as of 1/Feb - 1 point per dollar spent at participating merchants, 2 per dollar at Sears. Redemption moves away from the tiered system which could net you up to 3% on non-Sears and Sears purchases alike, to one that gives you a similar 1% return on non-Sears purchases.

It is unclear if the 1K -> 20K Petro conversion will remain, but if it does, it continues to be a great way to earn ~2.5 CX miles per dollar of gas purchased at PetroCanada.

EDIT: Note the T&Cs are different between the Sears Card Club and Sears Mastercard Club, and that the latter does not mention converting under redeeming points.