DMM Input Capacitance

[Kleinstein]

The measurement frequency can make quite a difference, as the input can have some resistance (and maybe inductance) from the protection in front of the main input capacitance. With a high test frequency the resistance can hide the capacitance, especially if large. With a low frequency series resistance is less relevant.

For the use of the DMM the additional resistance can also make a difference: it can be enough that loading an OP-amp is no longer an issue.

[KungFuJosh]

In practice, how often do you drop clipped wire ends and stuff into the mains and USB outlets? 

Never. The outlet is over a foot from my heat resistant work pad. However, if I start doing that now, I'm blaming you. 😉

[joeqsmith]

So you're injecting a MHz squarewave into the scope and then adding a bunch of stuff to the circuit (cables ...) and its having some effect.  Depending on edge rates, I can certainly understand this.  The fix may not be to find infinite impedance meters and reflectionless test leads.   

Depending what you have for probes, maybe you could flip the setup so the squarewave attaches to the meter using what ever the required termination is, then use a scope probe that fits what ever loading requirements you have.