Your bike definitely has cottered cranks. A crank cotter is not like a cotter pin. it is a round metal pin with threads on one end and the other end has a taper cut into it. The crank spindle has a flat cut into it just before the ends and when the pedal crank is pushed onto the spindle the cotter is pushed through the holes in the end of the crank arm and wedges against the flat on the spindle thus locking the crank arm in place. Despite appearances the nut threaded to the end is not there to be tightened to snug up the cotter s taper. It is there only to keep the cotter from falling out and needs only to be lightly tightened. When installed the cotter is tapped into place with a hammer until it is tight , then the nut is installed just over hand tight.

Since you are obviously a novice when it comes to cottered cranks you would be wise to take the bike to a shop , preferably one with an old timer for a mechanic, and have them remove the cotters for you. This will save you a lot of grief when you try to hammer out the old cotter and break the threaded part off and possibly spend many hours trying to get the remnants out.

As to your current problem, holding the opposing crank arm try wiggling each arm individually . also, hold the crank arm and pull up and down on it. Notice if
1 the spindle moves when you pull up and down or
2 the individual crank arm moves but the spindle( the shaft that goes from one side of the bottom bracket shell to the other ) stays in place.

If just the crank arm moves, take a ball peen hammer(not a carpenters hammer ) and tap smartly on the end of the cotter( not the threaded end ) a few times and check to see if the wiggling stopped. What you have done is driven the weged portion of the cotter against the flat on the crank spindle and tightened it. If this worked re tighten the nut on the other end of the cotter to keep the cotter in place .


If the crank arms are tight and the spindle is loose take a large pair of channel lock pliers and loosen the outer nut shown in your picture slightly then tighten the smaller nut very slightly , perhaps a 16th of a turn and snug up the outer nut, watching carefully to see if this tightens the inner nut also. If you get it too tight you will feel the drag and a grinding that indicates you are binding the bearings and things are too tight.

This is an adjustment that takes a bit of finesse. You want to eliminate the play without causing the bearings inside to get bound up. prepare to do this tightening and loosening a few times till you get it just right.