This statement is false and potentially dangerous.
Metal doesn't need to be a liquid to conduct electricity. The particles just need to be close enough for whatever voltage they are bridging to overcome the resistance provided by the binder. It's also possible for compounds to claim significant electrical resistance while still being easily capable of inducing enough stray capacitance to be problematic. Most thermal pastes are designed to conduct heat, not electricity, but there are certainly some thermal pastes that are flatly electrically conductive, and a huge number of pastes that are either capacitive enough to cause problems, or that can be rendered conductive through degradation or incomplete cleaning.
Stuff like
Aremco Heat-Away is advertised as a thermal grease, most of their their metal filled greases are designed to be highly electrically conductive and are. There are a thousand similar products, many of which make it to shops, like Amazon or AliExpress, rebranded as TIM, with little mention of how electrically conductive they are. It's often cheaper to reach a moderate level of thermal conductivity with an electrically conductive paste than with a paste with high electrical resistance.
This crap is almost certainly quite electrically conductive, as is
this...they are powdered copper and silicone oil.
Stuff like Arctic Silver has a high enough electrical resistance when fresh and mixed properly for them to get away with calling it non-conductive, but they still need to include the disclaimer about it's capacitance in the
product description, because if you do something silly like leave it in contact with a socket or closely placed SMDs, you could cause problems.
Beyond that, essentially every TIM that relies on metallic fillers, which is a large fraction of modern high performance pastes (pretty much any thermal paste advertising much more than 10W-m/K is either carbon filled, or metallic aluminum filled), can be turned electrically conductive, if enough binder is removed while leaving the filler behind.
The OP probably doesn't need to worry. As mentioned, the stock TIM is highly unlikely to be conductive or capacitive in any way, and most of the things any rational person would replace it with, other than liquid metal, should be safe enough for the location it will be used. However, is not even close to supporting the erroneous assertion that "conductive thermal paste does not exist".
No, generally not.
Both are under the broad category of Thermal Interface Materials...which is anything designed or used to conduct heat from one surface to another.
The subset of TIMs referred to as thermal pastes or thermal greases means some kind of thermally conductive filler suspended in a liquid binder. Paste and grease is usually interchangable here as most use some kind of oil as the binder, but I wouldn't consider the few non-oil based pastes to be greases. Most pastes using volitile binders are generally supposed to be dispensed on to a surface and then either heat cured or allowed to sit long enough for the binder to evaporate.
There are all kind of other TIMs, from metal shims, to compressible pads/gap fillers, to putties and gels. Liquid metal is exactly what it sounds like...a liquid alloy that is itself the thermally conductive material; there is no filler and no binder, just the metal.
The Upsiren LMTG-100 stuff that
@s1rrah mentions is a weird outlier. It's liquid metal blended into an oil binder...supposedly to increase the viscosity of the liquid metal and make it easier to handle, while keeping it's higher thermal conductivity. In reality this stuff seems to be the worst of both worlds, not the best.
