Whilst we're nit-picking terminology:
* The bottom bracket shell is the piece of the frame into which the down tube, seat tube and chain stays are affixed (brazed/soldered/glued/welded)
* The bottom bracket cups are the threaded pieces that screw into the bottom bracket shell. Usually the right hand cup is fixed and has a left hand thread to stop it being unwound by the action of pedalling. Usually the left hand cup is adjustable (winding it into the frame tightens the bearing) has a conventional right hand thread and has a lock ring around it. Pretty much every manufacturer likes to invent their own tooling for holding the cup and moving the lock ring.
* The bottom bracket axle is the steel/titanium bit to which the cranks attach, usually as a press fit with 4 tapered flats on each end. 2 nuts (in your case I think) or bolts stop the cranks from wandering off. (A popular misconception is that that nuts/bolts hold the cranks to the axle but let's not go down that rabbit hole just yet.)
* The bottom bracket bearing comprises the cups, axle and bearings which can be loose ball bearings, caged bearings or any variety of sealed needle/ball bearing cartridges.
* The average Joe's inability to grease/adjust a cup/cone/ball bearing bottom bracket plus the significantly faster fitting time has led to the almost total replacement of the bearings described above with the "modern" design.
I assume the 125mm relates to the length of the axle.
There is a great article on axle lengths/offsets here: https://www.renehersecycles.com/bottom-brackets-demystified/
Z
Edit: Link updated.
* The bottom bracket shell is the piece of the frame into which the down tube, seat tube and chain stays are affixed (brazed/soldered/glued/welded)
* The bottom bracket cups are the threaded pieces that screw into the bottom bracket shell. Usually the right hand cup is fixed and has a left hand thread to stop it being unwound by the action of pedalling. Usually the left hand cup is adjustable (winding it into the frame tightens the bearing) has a conventional right hand thread and has a lock ring around it. Pretty much every manufacturer likes to invent their own tooling for holding the cup and moving the lock ring.
* The bottom bracket axle is the steel/titanium bit to which the cranks attach, usually as a press fit with 4 tapered flats on each end. 2 nuts (in your case I think) or bolts stop the cranks from wandering off. (A popular misconception is that that nuts/bolts hold the cranks to the axle but let's not go down that rabbit hole just yet.)
* The bottom bracket bearing comprises the cups, axle and bearings which can be loose ball bearings, caged bearings or any variety of sealed needle/ball bearing cartridges.
* The average Joe's inability to grease/adjust a cup/cone/ball bearing bottom bracket plus the significantly faster fitting time has led to the almost total replacement of the bearings described above with the "modern" design.
I assume the 125mm relates to the length of the axle.
There is a great article on axle lengths/offsets here: https://www.renehersecycles.com/bottom-brackets-demystified/
Z
Edit: Link updated.