Rainsford does indeed set all three of the traps you mention in Richard Connell's short story, "The Most Dangerous Game." The Malay Man Catcher is apparently not a real trap but only a term invented by Connell for his story. Rainsford constructed the trap from two small trees:
His foot touched the protruding bough that was the trigger. Even as he touched it, the general sensed his danger and leaped back with the agility of an ape. But he was not quite quick enough; the dead tree, delicately adjusted to rest on the cut living one, crashed down and struck the general a glancing blow on the shoulder as it fell; but for his alertness, he must have been smashed beneath it.
The Burmese tiger pit is another Connell invention, though it may have some historical precedent under a different name. (Didn't I see this in the old Tarzan movies?) A hole is dug about four feet deep; sharpened poles are then embedded in the earth and placed with the sharp end up. The hole is then covered and camouflaged with the intent of surprising the victim who unknowingly walks upon the covering.
"...the pit grew deeper; when it was above his shoulders, he climbed out and from some hard saplings cut stakes and sharpened them to a fine point. These stakes he planted in the bottom of the pit with the points sticking up. With flying fingers he wove a rough carpet of weeds and branches and with it he covered the mouth of the pit. Then, wet with sweat and aching with tiredness, he crouched behind the stump of a lightning-charred tree.
The final trap, with his knife attached to a small tree, was tied to the ground. Another hair trigger was designed to trip the trap when stepped upon.
He thought of a native trick he had learned in Uganda. He slid down the tree. He caught hold of a springy young sapling and to it he fastened his hunting knife, with the blade pointing down the trail; with a bit of wild grapevine he tied back the sapling. Then he ran for his life. The hounds raised their voices as they hit the fresh scent. Rainsford knew now how an animal at bay feels.
It is a testament to the enduring popularity of Connell's story that these traps are now best known for their appearance in "The Most Dangerous Game."