If you look at most singer treadle cabinet, the tops are two pieces of wood screwed together, so the lift has a bend so it can be screwed to the lower section and the slant screws to the edge of the upper section.
If you look at this - see how the "top" has the main top, and then right around where the machine sits is a second layer of wood - that is what requires a "bent" base lift.
Here you can sort of see how it is mounted to the lower piece and the slant cut edge of the top piece. This is the style with the long spring instead of the short coil. This is also on a very old cabinet that a Singer D9 came in.
.
Here is the lift - showing the bent base.
This is the style with the short coil - which is more common
Cabinets for electric machines only have one layer of wood for the top - and the lifts have a flat base to attach to the bottom side of the cabinet top. These come in different styles, but the key part is the FLAT base.
BTW - this is the top that second lift belongs to - so it's a lot like the top of the desk you showed.
