How to time a linear motion exactly.

You are looking at the right two alternative ways of doing it. The first is using blends (old school), the second is using keyframes and that little stopwatch thing you clicked (new approach). You would do one or the other. Keyframe support is relatively new, so blends used to be the only solution.

First some theory. Puppets have a default value for properties. If you don't turn on the red button and adjust a property, you are adjusting the default value of the property for that scene. When you turn on the red dot for a property (but have not hit the record button yet) you are adjusting the starting value for the recording when you start recording. If you start recording then adjust the value, the recording will keep track of when you made the adjustment - it is recording what you do.

I suspect you started recording, then adjusted the Position X property. That is, the recording near the start contains the value of 0, then when it makes the jump that is where in your recording you changed it. I would delete that recording and start again. Instead,

• Turn on the red dot to arm the Position X value
• Drag the dragger to the final value you want
• Start recording for a period of time (a few seconds)
• Stop the recording
• You can now adjust the recording length by grabbing either end of the recording and stretching or trimming the length to whatever you need. The whole recording is the same final Position X value.
• Now do the blend like you did - it will go from the puppet default of 0 to the new value. 
• You might want to make the blend linear rather than ease-in and ease-out (straight line instead of curved line) - up to you. If fluid flowing down different sections of the pipe then I would not expect it to speed up and slow down.

But what if you want to start from a different value? Maybe change it several times? I would make a recording of the first value (arm property, set value to what you want, record a few seconds, extend/trim as required - in this case, extend to full length of scene). Don't do a blend. Then arm again, set it to the next value, record a few seconds, extend right edge out to end of scene, do the blend. Want a new value again? Arm, set initial value, record, extend the right edge of the tage again, do the blend. So you will end up with say 4 layers of recordings, each starting with a blend. What happens is the blend will go from the recording under it to the new value. So you can stack them up as many times as you need.

But that was the old approach. Keyframing might be better.

Instead, you click the stop watch icon next to the Postion X (like you did). That creates a little dot in the timeline and adds that property to the timeline. Drag the diamond to the start of the timeline. Select it and adjust Position X to whatever you want. Then drag the playhead (the blue vertical line) over to the end position. I don't have the software, I think you just change the Position X to the second value and it will create a new value (a new little diamond will appear). Move the playhead to the third position, change again.

Basically keyframing is for exactly what you want - the ability to say "at this time I want the value to be Z" and it blends automatically between them. You just move the little diamonds around to the time points you want. You can control the exact value, and change it as many times as you want (no need for multiple recordings).

But you would either do blends or keyframing, not both. I think keyframing makes more sense.