Are these circuits equivalent? (Arduino on breadboard > Atmega standalone)

Small background: I would print a PCB with an ATMEGA. Per Gammon website instruction (Gammon Forum : Electronics : Microprocessors : How to make an Arduino-compatible minimal board) we need 2 ceramic capacitors, on both "side" of Atmega, connected to the VCCs and GNDs.

I cannot figure if I can "delete" one because the circuit 1 will be equivalent to 2 or not.

Thank you very much

The capacitors should be located as close to the processor pins as possible, with a short path to ground.

They are not equivalent.

The Atmel datasheet recommends that the AVCC supply connection include an LC low pass filter to reduce the noise that is injected into the analog side of the processor.

The filter is shown in section 28 of the '328/P datasheet, page 313. It consists of a 10uh choke in series with the supply and the usual 100nf capacitor as close as physically possible to the processor pins.

Experience (of Arduino people in general) has shown that AVcc doesn't need the LC filter, just the capacitor, at least under typical arduino hobbyist conditions. Notice that the official arduino boards don't have it, and I've looked at dozens of '328p-based boards, and I don't think I've ever seen one that includes the filter.

But - you should still use the two ceramic caps, one on each side of the chip, to minimize the path length. Path length with decoupling caps really matters. You may be able to get away with one if the traces are really short, but it's better not to try. Ceramic caps are dirt cheap and simple, and the type of problems they prevent are the confusing, unpredictable kind that come and go seemingly at random and make you want to tear your hair out.

I don't think I've ever seen one that includes the filter.

The Pololu engineers always include the recommended inductor. Example

It is attempting to show that C2 and C3 should not just be placed on the board next to each other.

As in this photo (same website), C2 should be connected as close as possible to pins 20 and 22, and C3 as close as possible to pins 7 and 8. In this way, the bypass capacitors tend to block out outside inductive and capacitive pickup because the loop between them and the IC is so short. And fast current draw spikes caused by CMOS gate switching by the AVR have very little inductive line between the IC and the bypass capacitors.

These capacitors are also known as decoupling capacitors.

As far as the noise reducing inductor or resistor to AVCC, in general an Arduino layout has so little care given to noise reduction that it likely wouldn't make much difference. As much as it pains me to say that.

More information on decoupling the analog section power, AVCC, from the digital supply:
https://www.avrprogrammers.com/howto/avcc-aref

More information on bypass aka decoupling capacitors: