If your board had a PCF8574A mounted, the Slave addresses are 0x38 .. 0x3F
If your board had a PCF8574 mounted, the Slave addresses are 0x20 .. 0x27
Is that a real photo of the actual device on your desk?
In which case, it has a PCF8574 with address lines A0, A1, A2 "not shorted to GND"
So the Slave address is 0x27. e.g. your constructor would be
LiquidCrystal_I2C lcd(0x27, 2, 1, 0, 4, 5, 6, 7, 3, POSITIVE); // Set the LCD I2C address
Since you have spent a week of frustration, next time do a search for "I2C sniffer"
This sketch will diagnose which chip is on your adapter and what constructor you should use.
Yes, Adapters with PCF8574A do exist. They are not very common.
Please could you examine the chip markings on your Adapter.
David.