I read many conflicting accounts. The common belief seems to be "thermal noise increases with a larger resistance value, whereas current noise decreases when resistance increases."

But a web site mentioned this is just a myth (see below).

https://www.analog.com/en/resources/technical-articles/11-myths-about-analog-noise-analysis.html

"
8. The Amplifier with the Lowest Voltage Noise Is the Best Choice

"When choosing an op amp, the voltage noise is often the only noise specification considered by the designer. It is important not to overlook the current noise as well. Except in special cases such as input bias current compensation, the current noise is typically the shot noise of the input bias current: in = √2 × q × IB. The current noise is converted to a voltage via the source resistance, so when there is a large resistance in front of the amplifier input, the current noise can be a larger noise contributor than the voltage noise. "


So what kind of circuit where, as the resistance increases, the current noise decreases?
What kind of circuit where, as the resistance increases, the current noise also increases?

 

Current noise decreases as current decreases, which is not the same thing as saying that current noise decreases as resistance increases.

For any op-amp circuit, the overall noise is Gain*(√(vn^2+(R.in)^2) where vn and in are the voltage and current noises respectively.

Note that the voltage noise increases with the square root of resistance, but in many circuits the gain is proportional to a resistance, so doubling the resistance doubles the gain, but the noise only goes up by 41%.

Each situation must be taken on its merits, if you try to generalise, you'll make a mistake!

 

Current noise decreases as current decreases, which is not the same thing as saying that current noise decreases as resistance increases.

For any op-amp circuit, the overall noise is Gain*(√(vn^2+(R.in)^2) where vn and in are the voltage and current noises respectively.

Note that the voltage noise increases with the square root of resistance, but in many circuits the gain is proportional to a resistance, so doubling the resistance doubles the gain, but the noise only goes up by 41%.

Each situation must be taken on its merits, if you try to generalise, you'll make a mistake!

So why does it mention above that "The current noise is converted to a voltage via the source resistance, so when there is a large resistance in front of the amplifier input, the current noise can be a larger noise contributor than the voltage noise".

When there is large resistance, the current should decrease, so why should the current noise be increased from decreased current in the statement above?

 

So why does it mention above that "The current noise is converted to a voltage via the source resistance, so when there is a large resistance in front of the amplifier input, the current noise can be a larger noise contributor than the voltage noise".

When there is large resistance, the current should decrease, so why should the current noise be increased from decreased current in the statement above?

Because, in that case, the current is the bias current of the input transistors, which will be equal to the standing current in the long-tailed pair divided by hfe, so will be almost constant.
As I said, each case has to be taken on its merits or you will make mistakes. Make sure you use the right current.