Maybe a bit late for you, but I've dealt with this a lot.

As you've seen the exit tax gets charged when your ticket originates from Colombia. Round trips originating from elsewhere do not (in my experience) have this tax on the ticket.

The procedure to get the tax back is usually very easy, if you know what to do. At check-in, simply ask for a refund of the exit tax. Sometimes the agent will get confused and think you want the tourist refund of IVA on purchases so you need to correct them and say that this is for the timbre de salida. Maybe one time in four you'll need to ask for a supervisor.

The agent will verify your eligibility, then depending on the airline will either refund you cash on the spot (in pesos) or direct you to the sales office to fill out a form and get a refund processed on your credit card. AA and Copa will have you sign something and then refund the tax on the spot and in cash, Iberia will make you fill out a form at the sales office and then nag them relentlessly for a few months until they process it. For some reason every airline has a different procedure.

Very straightforward when you are on a direct flight out of the country. If you're on a connection you should still ask at check-in but they'll generally make you request the refund at your last stop out of the country. For example MDE-BOG-elsewhere on Avianca you go to the sales office in BOG to request--you'll need to wait in a line which you may not have time for unless you've got a nice long connection.

It can get fun if the ticketing, marketing and operating carriers aren't the same. Then everyone starts pointing their finger at everyone else. I recently flew MDE-BOG on Avianca, connecting to BOG-FRA on Lufthansa, all on an Avianca award ticket. Avianca check-in at MDE told me to talk to Lufthansa in BOG. Lufthansa check-in had me fill out a big long form to verify eligibility and then stamped it, then told me to go to Avianca to request the refund. After trying a couple of different desks at Avianca I got someone who called their supervisor and after a while told me to go back to Lufthansa for the refund. I then went to the aeronautica civil (the government entity administering the tax) where the two people working at the desk decided it would become their life's mission to get me the tax back. We went together to talk to an Avianca supervisor, then walked to the other side of the terminal to talk to the Lufthansa supervisor. Aeronautica civil determined it was Lufthansa's responsibility to refund the tax. Lufthansa said I needed to fill out their "contact us" form on their website, which of course isn't likely to get me anywhere. Then the aeronautica civil people took me to the superintendencia de transporte (another regulator) who told me that if Lufthansa doesn't respond to my request I should file a complaint with the superintendencia. They'll investigate, issue a fine to Lufthansa, and force them to give me my refund. It was shocking to see all of these government bureaucrats from two offices wanting to protect little old me from the big bad airline corporations.

I'm still waiting for a month to pass after my request to Lufthansa, then will file my complaint with the superintendencia and see what happens.