At least in London, the terms 'railways' and 'underground' owe a lot to the way the rail system was built in the UK- by private companies, and by a Parliament Act authorising each project. In short, right up to the 1930s, rail [and underground] operations were owned and run by Private Companies - with shares traded in the stock exchange.
In 1933, the 'Underground' system in London was nationalised, and under the new regime it expanded a bit- the Piccaddilly Line, Central Line and the consolidation of the District and Metropolitan Lines.
So right up to the end of the 20th centuary, you had the 'tube' [underground] in London run by London Transport, & the rail system run by British Rail.
Right now, Transport for London are responsible for running the tube system in an area called the Greater London Area-currently composed what are known as zones 1-6.
TfL gets a direct susbisty from central govn't to cover some of the costs- it was over £1.5b last year.
But the tube reaches destinations outside Greater London- such as
Watford, Epping , Amersham. Therefore you have higher ticket pricing
[zone 7,8,9- with much reduced element of support[.
However, TfL wishes to control all rail operation within their area, so they took over some traditional rail services- when all rail services were put to tender by the govn't.
This operation is known as overground. the main character is that the operation falls wholey within the Greater London Area.
The tube and the overground operation has the same ticketing and pricing per zone.
The rail companies operating into London have their own statuatory arrangements, different legal regime, and different pricing .
In practical terms, if you travel within the zone 1-6 boundary you pay the same pricing- but this took years to acheive and involved some complicated financial and legal arrangements between TfL and each of the various rail companies.
Crossrail is a different concept all together. This is a new rail line, which had to be enabled by an act of Parliament, authorising TfL to build and operate the line.
Crossrail would be run by TfL Rail- another sub company- with a fleet of 66 new trains.
Within London, it will charge the same zonal fares , but outside, it will have to charge different prices.
So, TfL runs three different rail operations:
The Underground, Overground and TfL Rail.
Paris is somewhat different, the Metro system was nationalised over 100 years ago, and was designed as a feeder to the rail network serving outer Paris and the suburbs.
The RER as a concept dates back to the 60s, but it is a copy from a much older idea- from Berlin.
In Berlin, there is the Undeground [U Bahn] and the S Bahn- which is the network of suburban rail services going into the heart of Berlin.
The services were always under the ownership of the City of Berlin [BVB and later BVG], and the rail system is fully compatible with DB.
So the Parisians simply copied the idea by creating the RER, cutting through the city centre.
I don't see any relevence to Israel; first of all, the rail network does not extend into the heart of Tel Aviv , Jerusalem or Haifa- i.e. there is no extensive underground system in existance.
and most importantly, Israel is too small for comparison- even if you take the whole country