MosfetRead Marks post.
Read the title at the top of the column, where itsays"Max".20V the max Vgs ,not the normal operating VgsThe max continuous drain current is given for a Vgs=10V.If it works to obtain the max current, it must therefore work for anything less than that.
Post a drawing with the MOSFET leads (S, D, G) identified. (With darker lines, please).
That MOSFET (if not damaged) should be fully saturated with 10V Vgs, make sure you have: Top of coil to 12V+, bottom of coil to Drain (look at 2nd picture), Source to 12V-, button from 12V+ to 220Ω resistor to Gate, don't forget Diode in parallel with coil with cathode to 12V+.