First airline, first type, either pilot can call 'stop' PF retained control and actioned.
First airline second type initially the same until about a year later when a crew had 'communication issues' and the call was misheard by PF so the call was changed to 'stop stop stop'. A year or so later it reverted back to just 'stop' for company commonality.
First airline, third type, as per type one.
Second airline, type one (same type as type three above) still the same.
Second airline, type two (a new type for the airline) The call was 'reject' and only by the captain but the actions were still by PF. Call changed to 'stop' about 18 months later (again for company commonality).
Third airline 'stop' call by captain only and actioned by the captain.

“ABORT” seems clearer to me. A clear and concise word.
However I am from pre Tenerife daze.

Only had one, in a helicopter coming off an offshore helideck just before rotation.
PNF “ABORT! Engine fire #2
Me: “ Aborting “

“Abort” was company SOP.

Yes it really was a fire. A small fire but a fire all the same. the fire and light went out when I chopped the throttle back on deck.

Strikes me the airlines (& managers) that you worked for must have had little confidence in the competence of their F/Os. In particular the third one: not allowing the F/O to call stop at any stage - and then only the Captain being allowed to action the RTO - seems to be bordering on downright dangerous. Perhaps that airline didn't allow the F/O to do anything! I well remember one of my first commercial flights as a second officer (but in the RHS) and being told "sit there, don't touch anything & speak only when you are spoken to"!! Fortunately things improved as CRM became a reality & the crusty old flying boat Captains retired......

First airline was "reject x 3" (I think?), called by either, done by PF, worked well every time. Current airline is type 3, and I don't like it at all. I have to take over control during a failure that might lead to control difficulty? FO as PF can continue flying and land after a failure, but not bring the aircraft to a stop, even though he does that every other leg???

Had an incident early in my FE career.
Lockheed Electra, the company procedure was for Captain to do the stop. FO was PF and as we started the take-off roll the captain saw an instrument problem and called STOP, STOP, STOP. The FO and I took our hands off the two sets of throttles but the captain did not take over and close them. We continued to accelerate so I reached forward and closed the throttles saying, "I'll stop the aircraft then".
Max speed was only about 40 kts.

Just to be clear, the F/O could call stop/stop stop stop/reject in all but the last two cases. The F/O, if PF, would remain in control for all except the last case.

Yet that is exactly the way both Boeing and Airbus recommend their aircraft are operated, a few operators (BA for instance) are very much outliers in the airline world by not following that recommendation.

What about “Cancel take-off clearance- STOP”

You enviously did not read the previous posts on this thread as to why not using those specific first words , ,just look at my poste #15 for instance.

At the risk of being considered frivolous, sometimes one needs to be quite emphatic.

I have no idea who the dramatis personae were, on this occasion, but the tale evidently related to an RAAF F111 during a reject at Amberly. The crew called, apparently most professionally, "Barrier, barrier, barrier" to which ATC evidently asked whether there was a problem. The crew's response was "(very naughty word) barrier !" this time several octaves higher and very much louder.

and did that helped ?
I also do not know what the correct military phraseology for a pilot to request an arresting barrier to be pulled. And Is there a standard "international " one , or is it different in each country ?

STOP, usefully, is a word widely recognised even by non English speakers. Most countries use it on road signs - even the French. (Though not of course the French Canadians.)

My recollection from some 50+ years ago is that "barrier, barrier, barrier" was the correct UK military phraseology back then. After a brief look at the latest online edition of the UK CAA Radiotelephony Manual (CAP 413) Chapter 10 (Military Specific Phraseology), that has not changed. I don't know if this is an international standard, but I suspect it is also used by most, if not all, NATO forces.

JT, et al, the alternative response; 'Barrier, barrier' - ATC lowers barrier !
Hence the need for standardised procedures relating to aircraft type before take off, those which are able to use the barrier, and those who cannot.
"When it's up it's up. When it's down it's down."

Re pedantic; there is increasing need for the industry use world-wide, internationally agreed practices via ICAO.

See the quick guide to ATC calls in the link below, it would be expected that the flight deck will use similar phrases. 'STOP'

N.B 'Cancel Takeoff' only given up to the point of commencing the takeoff roll.

https://skybrary.aero/sites/default/...kshelf/115.pdf

Also; https://contentzone.eurocontrol.int/phraseology/

Interesting , this is what the ICAO book (still) says : So for us the actual R/T to use was : [call sign] HOLD POSITION [repeated} , not : "Cancel take off" , as once again the words "take off" should only be used in conjunction with the actual take off clearance.

In Quebec we use WWT “Whiskey, Whiskey, Tango” AKA “Whoa, Whoa, Tabernac !!!!”….just joking.

Seriously In Quebec aviation wise “STOP” will produce the desired action.

I like the term ABORT because it is seldom used and therefore, when heard, gets your attention while “Stop” is often used ( at lesser volume ) in the cockpit and in radio communication.

When I first worked in France it was explained to me that “Stop” on a road sign was a command while “Arrete ” was used to indicate the location of a pull-off or Lay-bye, usually a scenic lookout where one rested, admired the view, enjoyed some old cheese, a fresh baguette and fine wine (Rouge in the north, rosé in the south) before getting back on the road.
The funniest was when I asked for car keys by asking “Les Clefs pour Le Char SVP” It was a military base…”You want the keys to a Tank!!???” The way they drive in the Var in the summer it would have been useful!

Many moons ago the main aircraft manufacturers and safety publications covered the notion of being..... "Go Minded" and to brief accordingly for departure.
ie. Is it better to reject in a heavy wide body and risk an over run or ......to take a problem into the air and manage it when time permits and safe ? Vmcg and V1 had to be addressed along with the use of reduced thrust take off thrust, which made every take off roll much longer. The call of "GO" was an option in the right scenario.