SSD not recognized by Windows 10 installer

Recently I've purchased a SSD (Crucial MX200) as replacement for the original HD of my laptop (Lenovo ideapad s510p).
The problem is that my bios doesn't recognize the SSD, even though the manufacturer (Crucial) guarantee the total compatibility of the device with my system (I checked this on their website).

I'll try now to summarize the situation:

• The SSD is detected by my current system (Windows 8.1 Home) or by a live distribution of Ubuntu, if connected through a USB 3.0 adapter.
• The firmware is up to date, version MU03, checked with "Crucial Storage Executive".
• The SSD is correctly recognized if installed into another laptop (I've succesfully installed Ubuntu 16.04 LTS on it just for trying).
• The SSD is not detected by the BIOS, Ubuntu installer or Windows 10 installer.
• The SSD is recognized by the BIOS if and only if at start up I press the hotkey for entering the BIOS and wait for 5-10 minutes (during this period the screen is black). If I reboot the system in order to run the Windows 10 (or ubuntu) installation, the SSD is not recognized anymore.
• The SSD doesn't show both in GParted or by using the diskpart utility of the Windows 10 installer.

I've tried every combination of BIOS's settings, like AHCI or IDE, UEFI or Legacy, disabling secure boot and so on. I've tried to disable everything I could, I've even tried to remove the DVD drive.
I've tried to format the SSD (while connected through the USB 3.0 adapter) both with Windows and Ubuntu, trying both MBR and GPT or leaving the space unallocated.

There is one thing that I've noticed. If I enter the BIOS (with the original HD inserted in the laptop), next to the name of the HD and the DVD drive there is an R, e.g.:
HD 500 GB ... (R1)
DVD .... (R2)

If instead I install the SSD and wait for the 5-10 minutes that I've mentioned before, the situation becomes like this:
SSD 250 GB ... (S1)
DVD .... (S2)

Basically, the R becomes S. I don't know the meaning of this, maybe it may be useful for you.

I've tried anything I could but I can't find the problem, any help/advice would be appreciated.

You may not be the only one with problems with that laptop...
SSD not recognized in new S510p - Lenovo Community

https://www.reddit.com/r/techsupport/comments/1rpmdn/lenovo_s510p_wont_boot_after_installing_intel_ssd/
Worked when cloned

Make and model should not make any difference as long as it's same type of connection.

Point is the idea- cloning. As to why your SSD isn't recognised, I don't know I'm afraid. Why cloning would succeed- no idea. Just a suggestion.

Needs someone with detailed knowledge of this area.

SATA controller in the laptop has a lot to do with disks. Can you see that SSD in the BIOS properly, like make and model ?

That is similar to this experience here on this laptop model from one of the links I posted:
05-03-2014 06:56 AM Hello Vincent,
you have tried to leave the PC on hold for 5 minutes or more?
After this time my pc goes into bios and the disk is recognized.
But .. unfortunately there is a but .. when I install Windows 8.1 does not recognize it ..
I do not think it's a problem of the discs crucial because I had a chance to try it on other PCs and they work very well.

And following that users' experience shows Samsung and Kingston SSD's work.

This again contrasts cloning vs swapping:
J'ai un nouvel ordinateur portable depuis 3 semaines (processeur i5 et windows 8.1) j'ai acheté il y a une dizaine de jour le M550 256Gb avec le kit de clonage (avec le logiciel Acronis True image HD), j'ai donc cloné le HD, tout s"est bien passé, mais quand j'interverti le disque dur et le SSD, mon ordi ne démarre plus

So it seems your PC and Crucial SSD may not make a happy couple...