How detection actually works
Every IP address belongs to an organisation, and that ownership is public. Address ranges registered to hosting companies, cloud providers and known proxy operators are catalogued, and any site can check an incoming address against those catalogues in a millisecond. That check is what produced the message you saw.
Some services go further and look at behaviour — how many different accounts have been seen from that address, whether it appears in abuse feeds, whether its reverse DNS looks like a server. But the first and most common signal is simply: this address belongs to a datacentre, and real customers do not browse from datacentres.
What does not fix it
Clearing cookies, switching browsers, using incognito mode, changing your user agent, disabling WebRTC — none of these change the address your request arrived from, so none of them change the verdict. Neither does switching to a different datacentre proxy, which is the mistake that wastes the most time: the new address is in the same category as the old one.
What does fix it
A residential address. These are addresses assigned by consumer internet providers to homes, so they are registered to an ISP rather than a hosting company and they carry the reputation of ordinary domestic traffic. A mobile address, assigned by a carrier to a SIM, is stronger still because carriers share addresses among many subscribers, which makes blanket blocking expensive for the site.
Check what your current setup looks like from outside before you change anything — the IP check tool shows the address and who owns it, which is the same information the site is looking at.
Testing that the fix worked
Send one request through the new address and read the result, rather than reloading the site and hoping:
curl -x http://USER:PASS@HOST:PORT https://api.ipify.org curl -x socks5h://USER:PASS@HOST:PORT https://api.ipify.org
If the address that comes back belongs to an internet provider rather than a hosting company, the detection you were hitting is gone. IP2World's residential pool carries both HTTP(S) and SOCKS5 on the same credentials, so either line above works.