It is just a cultural mistake. Every Doctor is a médico, but not every médico is a Doctor (Read below). In spite of they want to be called doctors... To be a Doctor you need to study much more than a grade career or specialitation, and for simplified need to "discover" or add something relevant to the profession . In the other hand "medicó" with accent in the "o" is a pass time and third person word, means he gave or put a medicine to some one ill, who in most of the case is in bed or a chair with difficults to move. And the last one medico with tacit accent in the "i" is the same of medicó just in present time but in first person. Yo medico a alguien. Nice word :)
Exactly you are acting like a nurse or medic, but you will not find the accent in the vowel "i" That is because it is a "palabra grave" it is mean that has accent in the penultimate syllabe but you don´t write this because the word ends with a vowel. The name for that thing is "Acento tácito"
I must say the audio doesn't pronounce it correctly, but it does not sound as "México". The D in "Médico" should be pronounce, but the audio says something like "Mé-ico", without a D. And the syllable "xi" in "México" is pronounced like a strong H sound, as in "hat", "hello" or "hate".
Medico y Doctor in Spain is the same. Cuando tell him Doctor My arm hurts for example but when you told about to go, you say I go to medico, sometimes I am going to the Doctor. Other think médica no is correct medico yes. Sorry my English. I'm Spanish learn English. August, 28 2019