TL;DR – free dynamic DNS with your own domain and cloudflare

• Enable port forwarding on your home router. You need to route port 80 outside to port 80 on your local server inside (if you want to use https -> Port 443)
• Buy a domain – search for cheap offers, but keep in mind that the domain register service needs to offer the option to add any DNS-server for name resolving your domain.
• Register at cloudflare, add your domain, chose the free plan and make cloudflare scan your domain DNS entries.
• Go back to your name register service (e.g. GoDaddy or Ionos) and add the cloudflare DNS servers as custom nameservers to your domain management console

wally.ns.cloudflare.com
oswald.ns.cloudflare.com

•  Get your cloudflare API key in cloudflare: Overview > Get your API key > Global API Key
•  Create a bash script on your local server, where you want to update your IP from and change the email, api-key, zone_record and record_name in the header of the script:

#!/bin/bash

# CHANGE THESE
auth_email="[email protected]"
auth_key="yourCloudFlareAPIKey"
zone_name="aboe.eu"
record_name="aboe.eu"

ip=$(curl -s http://ipv4.icanhazip.com)
echo $ip
ip_file="ip.txt"
id_file="cloudflare.ids"
log_file="cloudflare.log"

# LOGGER
log() {
    if [ "$1" ]; then
        echo -e "[$(date)] - $1" >> $log_file
    fi
}

# SCRIPT START
log "Check Initiated"

if [ -f $ip_file ]; then
    old_ip=$(cat $ip_file)
    if [ $ip == $old_ip ]; then
        echo "IP has not changed."
        exit 0
    fi
fi

if [ -f $id_file ] && [ $(wc -l $id_file | cut -d " " -f 1) == 2 ]; then
    zone_identifier=$(head -1 $id_file)
    record_identifier=$(tail -1 $id_file)
else
    zone_identifier=$(curl -s -X GET "https://api.cloudflare.com/client/v4/zones?name=$zone_name" -H "X-Auth-Email: $auth_email" -H "X-Auth-Key: $auth_key" -H "Content-Type: application/json" | grep -Po '(?<="id":")[^"]*' | head -1 )
    record_identifier=$(curl -s -X GET "https://api.cloudflare.com/client/v4/zones/$zone_identifier/dns_records?name=$record_name" -H "X-Auth-Email: $auth_email" -H "X-Auth-Key: $auth_key" -H "Content-Type: application/json"  | grep -Po '(?<="id":")[^"]*')
    echo "$zone_identifier" > $id_file
    echo "$record_identifier" >> $id_file
fi

update=$(curl -s -X PUT "https://api.cloudflare.com/client/v4/zones/$zone_identifier/dns_records/$record_identifier" -H "X-Auth-Email: $auth_email" -H "X-Auth-Key: $auth_key" -H "Content-Type: application/json" --data "{\"id\":\"$zone_identifier\",\"type\":\"A\",\"name\":\"$record_name\",\"content\":\"$ip\"}")

if [[ $update == *"\"success\":false"* ]]; then
    message="API UPDATE FAILED. DUMPING RESULTS:\n$update"
    log "$message"
    echo -e "$message"
    exit 1
else
    message="IP changed to: $ip"
    echo "$ip" > $ip_file
    log "$message"
    echo "$message"
fi

• Make the script executable and add it to the crontab

chmod +x updateIpCloudflare.sh
crontab -e
*/10 *    * * *   /home/pi/updateIpCloudflare.sh > /dev/null 2>&1