Problem solved friends! I received today an old 16k SIM card with the large chip still active, and the phone works properly without resetting itself. It makes and receive calls as it should, and the ringtone is maybe too loud.
I also have found a new-old stock battery for the phone. The seller tested it for me and not only accepted the first charge, but it lasted two days on his 9109 HA. Here are the first impressions from a brief usage of the phone:
-When you switch the phone on, it says "Self Test, Wait" and then "Please Wait". It does this a couple of times, before it finds network. When the phone detects the network and registers to it, on the first row appears the country code, usually in capital letters or in numbers in case the country code isn't registered in the firmware of the phone. On the second row of the display, there's the country code and the network code (in my case, on the first row there's I and on the second one "I 88", 88 is the code for Wind, an Italian mobile operator).
-The phone obviously doesn't have the CLI feature nor it's activatable, due to the age of the handset. Another particular thing is that it doesn't access to the phonebook of the SIM, so you can store your numbers only in the memory of the phone, quite a nuisance if (like me) you have a lot of numbers stored into the SIM. It also doesn't receive nor send SMS, although there's an envelope icon on the display, which probably is related to voice-mail I guess (I don't have the user manual of this phone)
-The audio quality is very good: the speaker and the buzzer are quite loud, and you can adjust the volume of the earpiece by pressing Vol + or - during a call. The ringtone volume can be adjusted in the menu. Like all the GSM phones of the early '90s, you can't change the ringtone nor you can edit the greeting message. However, a curious thing is that when you turn it off, the phone displays "Bye" for a very short time, before switching off. Funny!
-The phone was conceived to be a direct competitor of the analogue Motorola Microtac 9800x and Classic series. Both of them, like the Alcatel 9109 HA, had the microphone integrated into the flip rather than the cellphone body, but unlike the Alcatel, they had a more advanced menu and a large choose of ringtones. On the other hand, the 9109 HA allows you to put a password for the menu, along with the PIN code and the phone lock code.
-The LCD display behaves like an LED one. Why? Because this is probably one of the most singular things I have noticed on this cellphone. Basically, you can view the display only when the backlight is on. After the backlight turns off, the display disappears as well. In order to see text again, you must press a random key and the backlight (along with the display) will turn on. It also turns on when you receive a call. The green LED located above the display is permanently on, in order to show that the phone is up and running. So we can say that the display of the 9109 HA, despite is a LCD, behaves exactly like an LED one. I'd expect to see this behavior on a Dynatac or a Microtac with an LED display, rather than on a cellphone with an LCD one. But these little quirks make French cellphones so lovely!
I will keep fiddling with the phone in the next days, and I'll use it as my everyday mobile for a while, to see if it's a good phone to use on daily basis, like the GH172, and I'll post more impressions here.