"Wednesday week"

"Wednesday week" is normal British English, and you have interpreted the meaning correctly. There's also "fortnight" used in exactly the same way, although possibly more rarely ("Wednesday fortnight" is two weeks from Wednesday).

I'm not aware of a particular name for this idiom.

British English does not use "today morning". It would be "this morning". I've never heard "today morning," so I'd have to guess it was either Indian-English or a personal idiosyncrasy.

I've heard Wednesday week and Wednesday fortnight used, mainly by English people. English speakers in Wales would be more likely to reverse this to 'a week Wednesday' or 'a fortnight Wednesday'. For example 'we will meet again a fortnight Wednesday'.

Wednesday week is common in the South East US as well. It can apply to any day of the week. Monday week is common for the following Monday.

I have several friends from India who will also use the "today's morning". I believe it is an Indian-English idiom.