I have a doubt about the intuition behind Electric field and Potential difference. Let's imagine a closed circuit with a battery, a resistance and a lightbulb. We observe that the lightbulb is on. What I infer from this is that electrons are slowly going around, and thus there must be an electric field. It seems intuitive to think that to keep the electrons moving, the electric field must act everywhere in the circuit (1).
However, if I measure with a voltmeter the voltage difference at two points in an ideal wire, without no components in between, the voltage difference is zero, which by the definition of electric field and voltage implies that the electric field is also zero (2).
Point 1 and point 2 are contradicting one another. What is wrong here?