at which point does an officer have to read you your rights after being placed under arrest for dui

Many folks wrongly but reasonably believe that a police officer must immediately inform an arrested person that he or she has the right to remain silent and other rights.  They are referring to "Miranda" rights, or warnings, which in fact must be provided to an individual by a police officer if two conditions exist: First, the individual must be "in custody."  That doesn't necessarily require handcuffing or the actual restraint of one's physical liberty.  A person is "in custody" if, given all of the attendant circumstances, a reasonable person in the given situation would believe that he or she is not free to leave the presence of the officer.  Secondly, the officer must intend to ask questions of the person who is "in custody." Standard procedure by police in DUI investigations is to not provide Miranda warnings until after "non-testimonial" aspects of the investigation have occurred.  Those would include roadside sobriety tasking and breath, blood or urine gathering.  That is because they know that Miranda warnings may cause an arrested person to not cooperate with that part of the investigation. If a person, without Miranda warnings being provided, at any time during any investigation, states to a police officer that he or she wants to consult with a lawyer, or is unwilling to answer any questions, there can be no interrogation.   So, during the investigation when you mentioned your desire to speak to an attorney, you were then invoking your Constitutional rights to an attorney and to silence.