It's an old car, and I just had a local mechanical shop serviced the car, including its battery.
After the service, I started noticing difficulties at engine start, with clicking noises. I can reliably reproduce this symptom, and to avoid the issue, I have to use force and turn the key very hard to the end to start the engine. Previously, I never had to use so much force to turn the key.
Please let me know if you need more information, and I highly appreciate any hints and suggestions.