Abul - it's important to separate the legal and beneficial ownerships when dealing with land/property and specifically when one of joint owners dies.
The guidance you have copied from relates to their beneficial ownerships. That can be split and relates to the 'share' you mention. You can split/share your beneficial ownerships but the legal ownership which we register is always dealt with as a whole. It can't be spit/shared.
To put it into your terms your Father's share can be passed on/left to anyone but that's his beneficial share, the £s and pence value of his share in the property.
But the legal ownership, the land, bricks and mortar, can't be split/shared so that passes as a whole to the surviving joint owner, your Mother.
So when your Father died IF he left his beneficial share to you then you and your Mother have to decide how you wish to deal with that. You have options to consider and the two most common are either just updating the register re the death or her transferring the legal ownership from herself to herself plus you.
My advice would be to seek legal/financial advice to decide what options exist and what's best for the two of you - things to consider are whether the 'tenants in common' aspect is sufficient to protect your beneficial interest, whether being a legal owner impacts on your own property ownership in some way or impacts when you may buy your first home etc
Hope that helps explain the difference between the legal ownership, which is what your first enquiry was about and the beneficial ownerships/TIC that you have then added
NB - there are a number of threads on the forum that deal with similar enquiries and the nuances between legal and beneficial ownership. For example this one may assist -
Death of a tenant in common · HM Land Registry