The steel inner hose should stay inside the hose, when breaking it should rest against the barrel adjuster (top-left)

If the inner cable hose is able to push through the barrel, it means the hole in it is too large (or the cable and hose too thin!) but also that the hose is really shitty quality for breaking like this

I'd recommend starting with buying the cable hose somewhere else, and maybe check the barrel adjuster as well, it should only allow the cable through

You're not doing it wrong, cables corrode/wear out. If the bike is that old and rusty, replacing the cables is normal, as is the tires/tubes, the brake pads, spokes (they might snap if you're truing up the wheels), chain links (if any are corroded), basically all of the wearable non-frame stuff. If there is rust on the frame, it might be structurally compromised and unsafe to ride.

That's crazy. I've only ever seen this happen after an emergency stop with compressionless housing while not using a ferrule at a|l cable stops. 😄

The sibling comment has the solutions.

You'll need a correctly sized stopper cap like the metal end cap on the black housing in your pic. Though clearly that one has been blown through. The opening in it should be just large enough to let the break cable through. You can add an external one onto this housing if you find one

Replace the housing with higher quallity housing. You need a piece for those brake levers that is missing. It is like a step down ferrule. Something like this. You are going to have to go on a journey to find it. If searching locally find the place that loves to work on older style bikes. They got old shit everywhere. You need stiff as hell housing but not shifting housing. Basicaly modern brake housing. So two issues. Wrong housing with no stepdown ferrule. A barrel adjuster for a brake will not fit and a shifting barrel adjuster will just break.