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WindScribe was the issue as it was filtering the Captcha. When using a different VPN provider the Captcha loads correctly. I found this out after I came back from the trip, I'm leaving my other findings below for reference.

I got a physical card from the desk and loaded the weekly subscription onto it. This cannot be purchased from the machine, as you need a personalized card.

Troubleshooting:

When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference, this is how the page of step 5. of the question is supposed to look like:

WindScribe was the issue as it was filtering the Captcha. When using a different VPN provider the Captcha loads correctly. 

I found this out after I came back from the trip, I'm leaving my other findings below for reference.

I got a physical card from the desk and loaded the weekly subscription onto it. This cannot be purchased from the machine, as you need a personalized card.

Note of caution: two people in our group managed to load a ticket on their phone. One had no issues, the other had problems after two days. The ticket was suddenly not present anymore, and reloading it caused an error. We went to several service desks, but none of them was able to solve this issue. This meant that for the remaining 5 days, they had no valid ticket even though they paid for it. It might be worth it to avoid all this by getting a physical ticket (extra cost of €2).

Troubleshooting:

When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference, this is how the page of step 5. of the question is supposed to look like:

It seems like the Captcha was the issue, but I couldn't determine what was causing the Captcha to fail. As a solution I got a physical card from the desk and loaded the weekly subscription onto it. This cannot be purchased from the machine, as you need a personalized card.

I think it's due to the VPN service I was using (WindScribe) When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference of others, this is how it's supposed to look like:

WindScribe was the issue as it was filtering the Captcha. When using a different VPN provider the Captcha loads correctly. I found this out after I came back from the trip, I'm leaving my other findings below for reference. 

I got a physical card from the desk and loaded the weekly subscription onto it. This cannot be purchased from the machine, as you need a personalized card.

Troubleshooting:

When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference, this is how the page of step 5. of the question is supposed to look like:

It seems like the Captcha was the issue, but I couldn't determine what was causing the Captcha to fail. As a solution I got a physical card and loaded the weekly subscription onto it.

I think it's due to the VPN service I was using (WindScribe) When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference of others, this is how it's supposed to look like:

It seems like the Captcha was the issue, but I couldn't determine what was causing the Captcha to fail. As a solution I got a physical card from the desk and loaded the weekly subscription onto it. This cannot be purchased from the machine, as you need a personalized card.

I think it's due to the VPN service I was using (WindScribe) When I got to France, I tried to open the account registration page, but it still threw the Error 15 page. The same error was thrown when I was connected to the EuroStar wifi. Only when I was connected to the wifi of my Airbnb was I able to get to the registration page again and the Captcha loaded.

For future reference of others, this is how it's supposed to look like: