Just out of curiosity, how can you tell?
It is usually mentioned on the ticket(s), but different countries do it in a different way.
in Germany (and adjacent countries, except France) you normally get a single ticket for the whole route. My sister for example bought a ticket from Brussels to Wengen, and it mentions this:
Hinfahrt Bruxelles Zone Wengen
Via: ACKLM(FMA/MZ)KAOGBAS OltenBernInterlaken
OstLauterbrunnen
So this ticket is valid from any station in the "Zone Brussels", which is about 30 of them, and allows you to travel to Wengen via Aachen, Koeln, Limburg, then either via Frankfurt and Mannheim or Mainz, then Karlsruhe, Offenburg, Basel, where you change to the Swiss network (that is what the 1185 is) and then Swiss Rules apply. (SBB does not use abbreviations for the route like DB does).
But it also mentions:
Zugbindung ICE 11, 06:23 Uhr ICE 103, 08:55 Uhr
So you normally have to take ICE 11 and IC 103, the other parts of the route are free.
Important is that you do not miss the first train you booked. From then you just take trains in a logical succession. As soon as there is a delay that means you will arrive more than 20minute lates at your destination (which is usually the case as soon as you miss a connection) the "Zugbindung" is cancelled. Then you can take any train on the route mentioned under VIA: (but in practice you can just take whatever the on line planner suggests as an alternative).
But when the trains are on time you have to take in this case ICE 11 and then connect to ICE 103. After that you are free to take whatever you want.
Further down on the ticket the schedule is mentioned for the whole trip, but that does not mean that you have to take those trains. You have to realise that it is the ticket on the top of the page, and the QR code that matters. The rest is just filler.
I read in some other websites (note plural), when your train is late,
you need to get proof from the conductor otherwise you can get fined
for jumping onto the next train.
That depends really on the network. In Germany and Switzerland you do not need to do anything. Train conductors can check that the story you tell them is correct. As an example: Last year I travelled Vienna - Interlaken using the NightJet (booked with OBB) and then a connecting train for which I had a SBB Supersaver ticket. NightJet was late, and I missed the train I was supposed to take, but it was no issue, as the conductor on the train to Interlaken already knew that the NJ had arrived late in Zurich.
(And this was on two separate tickets).
I am doing this trip again next week.
In other countries things may be different. For example in Italy if you book a multiple train ticket you will get multiple tickets. So for example if you book Spiez - Rimini with a change in Milan you get a ticket Spiez - Milan and another ticket Milan - Rimini. If your train is late to Milan you need to go to the Frecciadesk at the head of the platforms, and they will just exchange your Milan - Rimini ticket for one on the next train. Don't go to the ticket office, as that is often very busy...
In France you just aks the conductor, and he will scrible something on your ticket (btw, that is why even when I book tickets online I always print them out, so the conductor has something to scrible on...) and then you just indeed hop on the next available train, and ask that trains conductor to find a free seat for you. If that is not possible you can just sit on one of the fold-out seats in the vestibule...
How they do it in Spain I don't know. Last time I took a train in Spain the railways did not sell connecting tickets. I remember standing in a station in Galicia and asking for a ticket to Barcelona, and not getting one. "There are not trains to Barcelona from here..." .