I've been doing some experiments with this myself - I decided to buy a Toshiba T1910CS with a bad front polariser thinking it would be simple to fix. Polarisation is unfortunately a rather complex topic - linear polarisation is all that's required with a TFT, but with passive matrix DSTN panels it seems to me that there's more to it. I haven't been able to find good discussions on english language forums for computer type polarisers and mostly when I see polariser discussions it usually just ends with 'replace the screen' - sadly that's not possible with many of these laptops, your Toshiba 2140CDS uses an HPA DSTN panel that is probably the only panel that can work with the signals being sent to it.
On mine it's a Toshiba TLX-8101S-C3X that is essentially unobtainable and DSTN panels this size generally aren't available, so I've put the time into cleaning off the polariser from mine. It did some interesting damage - the polariser breaks down into acetic acid / vinegar type stuff hence the name 'vinegar syndrome' and on mine, the vinegar caused surface rust all around the front edges of the panel.
I have a similar problem to you in that a regular polariser doesn't give a clear or colourful picture (except inverted which is useless), at any angle I've tried either front-ways or back-ways. The ones I've tried so far are linear polarisers and I can compare them with the remaining segments of a damaged original DSTN polariser - which looks great with nice black levels and bright colours.
I recommend trying out some reald-3d glasses to test polarisation:
They have circular polarisers on each lens, one is clockwise and the other is counter-clockwise. A circular polariser is an additional 'quarter wave plate' layer that as far as I can tell makes the light polarisation effect into one-way, beyond just linear polarisation. If light goes through a circular polariser (circular + linear polariser) then to work with it, the polariser on the other side needs to have a matching circular polarisation. I do not know the proper terminology for this yet.
These reald-3d glasses have different direction circular polarisers and I found that the left eye lens gives a good picture. According to here that's a counterclockwise circular polariser: https://www.diyaudioandvideo.com/TV/3D/Theater/
Here are some examples of the different types of polariser, some re-used offcuts of linear polarisers from broken TFT screens, the two lenses of some reald-3d glasses and the original polariser remains:
I haven't yet found a good replacement, I tried cold-light OLED polarisers for an iPad but the circular polarisation seems to be the wrong way or on the wrong side from the glue. Consider trying this place but they don't say which direction the circular polarisation goes: https://3dlens.com/lcd-polarizer-film.php
Beyond that, I think trying out different types of oled polarisers might work if it's possible to get one that goes the other way, mine were meant for replacing the front polariser of an ipad pro screen, perhaps there are ones for the other side. It seems that beyond clockwise and counterclockwise, for purchasing they can be referred to as left-handed or right-handed. No idea which is which yet.
Right now - those glasses with lenses from 2x reald-3d glasses are required to use the laptop 😁