Try installing the active LTS version instead of latest with nvm install 16 (instead of nvm install node) as per the examples.

This was tested on the docker image amazonlinux:2 and correlates to what AWS CDK supports (ref)

Update

LTS has moved on for node, however, as per @djvg's comment, note there is a limitation with amazon linux 2 on glibc, so if requiring node > 16 will need to move to newer amazon linux version, eg: https://github.com/amazonlinux/amazon-linux-2023/blob/main/Release-Notes-Amazon-Linux-2022.0.20220728.0.md

The answer at:
Node.js 18 is compiled for glibc 2.28
is certainly correct although highly unsatisfying.

You cannot run the binary on your system because it relies on a version of glibc which is not available on your system.

$ docker run -it amazoncorretto:11 bash
bash-4.2# cat /etc/os-release
NAME="Amazon Linux"
VERSION="2"
ID="amzn"
ID_LIKE="centos rhel fedora"
VERSION_ID="2"
PRETTY_NAME="Amazon Linux 2"
ANSI_COLOR="0;33"
CPE_NAME="cpe:2.3:o:amazon:amazon_linux:2"
HOME_URL="https://amazonlinux.com/"
bash-4.2# rpm -q --provides glibc|grep -i libc.so
libc.so.6()(64bit)
libc.so.6(GLIBC_2.10)(64bit)
libc.so.6(GLIBC_2.11)(64bit)
libc.so.6(GLIBC_2.12)(64bit)
libc.so.6(GLIBC_2.13)(64bit)
libc.so.6(GLIBC_2.14)(64bit)
libc.so.6(GLIBC_2.15)(64bit)
libc.so.6(GLIBC_2.16)(64bit)
libc.so.6(GLIBC_2.17)(64bit)
libc.so.6(GLIBC_2.18)(64bit)
libc.so.6(GLIBC_2.2.5)(64bit)
libc.so.6(GLIBC_2.2.6)(64bit)
libc.so.6(GLIBC_2.22)(64bit)
libc.so.6(GLIBC_2.23)(64bit)
libc.so.6(GLIBC_2.24)(64bit)
libc.so.6(GLIBC_2.25)(64bit)
libc.so.6(GLIBC_2.26)(64bit)
libc.so.6(GLIBC_2.3)(64bit)
libc.so.6(GLIBC_2.3.2)(64bit)
libc.so.6(GLIBC_2.3.3)(64bit)
libc.so.6(GLIBC_2.3.4)(64bit)
libc.so.6(GLIBC_2.4)(64bit)
libc.so.6(GLIBC_2.5)(64bit)
libc.so.6(GLIBC_2.6)(64bit)
libc.so.6(GLIBC_2.7)(64bit)
libc.so.6(GLIBC_2.8)(64bit)
libc.so.6(GLIBC_2.9)(64bit)

It supports only up to GLIBC_2.26.

Also amazoncorretto:latest only supports up to glibc 2.26:

$ docker run -it amazoncorretto bash
bash-4.2# cat /etc/os-release
NAME="Amazon Linux"
VERSION="2"
ID="amzn"
ID_LIKE="centos rhel fedora"
VERSION_ID="2"
PRETTY_NAME="Amazon Linux 2"
ANSI_COLOR="0;33"
CPE_NAME="cpe:2.3:o:amazon:amazon_linux:2"
HOME_URL="https://amazonlinux.com/"
bash-4.2# rpm -q --provides glibc|grep -i libc.so
libc.so.6()(64bit)
libc.so.6(GLIBC_2.10)(64bit)
libc.so.6(GLIBC_2.11)(64bit)
libc.so.6(GLIBC_2.12)(64bit)
libc.so.6(GLIBC_2.13)(64bit)
libc.so.6(GLIBC_2.14)(64bit)
libc.so.6(GLIBC_2.15)(64bit)
libc.so.6(GLIBC_2.16)(64bit)
libc.so.6(GLIBC_2.17)(64bit)
libc.so.6(GLIBC_2.18)(64bit)
libc.so.6(GLIBC_2.2.5)(64bit)
libc.so.6(GLIBC_2.2.6)(64bit)
libc.so.6(GLIBC_2.22)(64bit)
libc.so.6(GLIBC_2.23)(64bit)
libc.so.6(GLIBC_2.24)(64bit)
libc.so.6(GLIBC_2.25)(64bit)
libc.so.6(GLIBC_2.26)(64bit)
libc.so.6(GLIBC_2.3)(64bit)
libc.so.6(GLIBC_2.3.2)(64bit)
libc.so.6(GLIBC_2.3.3)(64bit)
libc.so.6(GLIBC_2.3.4)(64bit)
libc.so.6(GLIBC_2.4)(64bit)
libc.so.6(GLIBC_2.5)(64bit)
libc.so.6(GLIBC_2.6)(64bit)
libc.so.6(GLIBC_2.7)(64bit)
libc.so.6(GLIBC_2.8)(64bit)
libc.so.6(GLIBC_2.9)(64bit)

You cannot update your glibc library because it is part of the Linux Kernel. It will be upgraded when you migrate to a newer Operating System Version.

The way to get it running on your system would be to rebuild it from the Source Code for your system and ideally package it for easily reproducible installation.

You might setup a docker image from amazoncorretto Base Image with the Packaging Environment set up.

An Package build on the Packaging Docker Host will be installable on all amazoncorretto:11 Systems.

Useful will be the pre-built .spec file which can be found at:
NodeJS .spec file
Then you only need to build your rpmbuild environment to create your installable NodeJS .rpm package.

AWS NodeJS Lambda Runner:

@Rich mentioned the public.ecr.aws/lambda/nodejs:18 AWS Lambda Runner image.
This is an image which is designed to run NodeJS tasks with the CMD dockerfile instruction at startup.
This actually might be fine for some single shot task (what it is designed for) or to startup directly a ExpressJS web app.
A downside is that you cannot independently update the NodeJS engine without upgrading the whole base image because the NodeJS engine is not installed as a system package.

$ docker run -it public.ecr.aws/lambda/nodejs:18 
19 Jun 2023 09:53:51,148 [INFO] (rapid) exec '/var/runtime/bootstrap' (cwd=/var/task, handler=)


^C19 Jun 2023 09:54:26,776 [INFO] (rapid) Received signal signal=interrupt
19 Jun 2023 09:54:26,776 [INFO] (rapid) Shutting down...
19 Jun 2023 09:54:26,776 [WARNING] (rapid) Reset initiated: SandboxTerminated
19 Jun 2023 09:54:26,776 [INFO] (rapid) Stopping runtime domain
19 Jun 2023 09:54:26,776 [INFO] (rapid) Waiting for runtime domain processes termination
19 Jun 2023 09:54:26,776 [INFO] (rapid) Stopping operator domain
19 Jun 2023 09:54:26,776 [INFO] (rapid) Starting runtime domain

$ docker run --entrypoint bash -it public.ecr.aws/lambda/nodejs:18
bash-4.2# rpm -qa|grep -i node|wc -l
0
bash-4.2# node --version
v18.16.0
bash-4.2# npm --version
9.5.1
bash-4.2# cat /lambda-entrypoint.sh
#!/bin/sh
# Copyright 2020 Amazon.com, Inc. or its affiliates. All Rights Reserved.

if [ $# -ne 1 ]; then
  echo "entrypoint requires the handler name to be the first argument" 1>&2
  exit 142
fi
export _HANDLER="$1"

RUNTIME_ENTRYPOINT=/var/runtime/bootstrap
if [ -z "${AWS_LAMBDA_RUNTIME_API}" ]; then
  exec /usr/local/bin/aws-lambda-rie $RUNTIME_ENTRYPOINT
else
  exec $RUNTIME_ENTRYPOINT
fi
bash-4.2# ls -lah /usr/local/bin/aws-lambda-rie
-rwxr-xr-x 1 root root 5.6M Jun 12 13:21 /usr/local/bin/aws-lambda-rie
bash-4.2# ls -lah /etc/yum.repos.d
total 12K
drwxr-xr-x 2 root root 4.0K Jun 23  2022 .
drwxr-xr-x 1 root root 4.0K Jun 19 09:57 ..
-rw-r--r-- 1 root root 1003 Oct 26  2021 amzn2-core.repo