The connection diagram on the upper left of your picture looks like a normal cross-over (null-modem) serial cable for DB-25 connectors.
Compare with the pinout table on e.g. Wikipedia. What we have is:
DB-25 serial crossover
1 --- 1 ("Frame" / likely the shielding around the cable)
2 --- 3 Tx -> Rx
3 --- 2 Rx -> Tx (data lines crossed)
4 --- 5 RTS -> CTS
5 --- 4 CTS -> RTS (control lines crossed)
6 --- 20 DSR -> DTR
20 --- 6 DTR -> DSR (another set of control lines crossed)
7 --- 7 GND
8 --- 8 DCD
But if you look at the pinout table for the smaller DE-9 connector, you'll see that the pins are not in the same positions. (They can't, obviously, since pins 10-25 don't exist.) For example, there's no separate pin for the protective ground, and 9 is the ring indicator.
If you need to make a crossover cable from DB-25 to DE-9, you'll either have to take into account the different pinouts, or acquire a DB-25<>DE-9 adapter, and then make/acquire a DE-9 crossover cable. Just make sure to find one with all the control lines connected (and you may need to check that they are not connected oddly, there are different variants on how DTR, DSR and DCD relate to each other.)
Of course, there are tables for the DB-25<>DE-9 crossover connection, e.g. here. That one has DCD combined with DSR, but in your diagram, they're separate.