Measuring a magnetic pulse using a coil, and circuit adjustments
When you have a fast rise time in current or voltage, the inductance of all current loops and load capacitance are both important and often overlooked. Also the ratio of that reactive impedance to resistive at its resonant frequency determines the dampening factor or its inverse gain ratio, Q .  In this case I see 25MHz with high Q >10.

Model errors

There will always be some capacitance in every part connected to or near the ground or supply.  This includes the FET and your scope probe and finger.

This is when you need to add stray inductance and FET  capacitance to your schematic model. (See specs for Ciss and Coss) . For instance a 1cm wire or resistor to the gate can be 10nH and the Ciss is 5150 pF which resonates near the 25 MHz of your gate Vgs pulse.  This has a resonant impedance around 1.5 Ohms so 0.1 Ohms results in a Q >10, so increase R3 to at least the Rg of the FET = 1.5 Ohm for a more critically damped response.

Then all coils have interwinding capacitance, so all Inductors have a unique Self Resonant Frequency (SRF)  parallel for L and in series for Capacitors with some inductance and often referred as Series Resonant Frequency, because microwave Caps also have a PRF (parallel mode).

Probe effects

Even a scope probe with low 60 pF/m coax will resonate with your probe ground wire inductance, ESL of ~10nH/cm so that inductance must be kept short as well.

Then there are stray effects which are much smaller in your layout such as coupling capacitance, Cs, to ground effects and series inductance to load, ESL.  All 10:1 probes are notorious for >20MHz resonance with long ground clips on a pulse wave.

Finger results

Since the FET inverts and your finger has high capacitance over a resistor area and your body conducts and radiates this putting your finger on the gate R3 tends to couple to the radiated resonant output field and cancel. It also reduces the rise time of the gate signal.

Putting your finger on R4 acts as a bypass cap to the stray inductance of that circuit which is part of the same current loop as the ground signal. So that too bypasses some current bandwidth thru your finger instead of the inductive resistor path. R4= 1 Ohm is equivalent to about 10nF at 25MHz which is what you might get by pressing hard on a resistor.

Calibrating your fingers on an RLC meter is a good way to understanding how to tune you circuit.  

Anecdote

When I was young, there were several times when I wished I could ship my finger along with my final instrument design ... ;) 

That is until I mastered EMC by a thorough understanding of RF loops, stray capacitance and slot/strip antennae.  Then you can estimate the radiated noise and susceptibility issues with resonance just by looking at the geometry of the layout and the bandwidth of the signal.  Estimating tracks by RLC and Z. (Hint: get an RLC Nomograph and Saturn PCB analyzer software.)  Examine the effects of ESD @ 0 to 300MHz which depends on the gap of the avalanche effect. (arc in air)