Can I guess 5/3-wire RS-232 pins from voltages?
First all, you shouldn't need a MAX232 based TTL logic shifter for this setup, as you are already dealing with logic level signals (0 to 5V).  I assume your Arduino is also 5V; otherwise you will need to do level shifting between 5V and 3.3V.  There are lots of posts already on this forum re that so I won't repeat them (search for level shifter 5v 3.3).

I assume you are looking for the labels as they come out of the connector, i.e. TXD means the pump is sending the Arduino and connected to it's UART RXD pin.  Vice-versa for RXD (transmit from Arduino TXD to pump).

I would try:  3 is TXD, 1 is RXD and 5 is GND.  I would put a 330 ohm resistor (or similar value) in the RXD line just in case I'm wrong. Even if shorted, this will limit the maximum current to 15 mA, which shouldn't harm your Arduino.

After connecting up the line to the Arduino, both sides of the resistor should be close to 5v (instead of the 1.26v) since the RXD input should be drawing very little current.

If pin 1 doesn't work for RXD, try pin 2 or 4 (again with the resistor).