> I am not familiar with the device names in the power cabinet
"Fuses".
In fact they have not been fuses in a long time. Circuit Breakers arrived in 1922 and became common in new-work by 1960. But a circuit-breaker is (mostly) a resettable fuse.
In the UK they have a LOT of very strange names for the things that they put where fuses/breakers go. The overload-blowout function is almost incidental to the additional protections available. You probably have some of these. Names like Residual Current, only abbreviated.
> rated at 10amps, is that really the maximum of its loading capabilities, or is it like 15 amps but the companies just say its 10 as a safety margin?
Does it matter what they say? Nobody (except a few here) could ever add-up the actual load.
"Maximum"? Wires and fuses don't drop dead instantly on 101% load.
"In General"..... the electric business picks a reasonable size of load for one circuit. A couple thousand Watts is good. Much smaller means too many circuits, too much labor. Much larger, and the whole house goes dark when one fuse pops.
With the local voltage, this gives the Amps.
They design a fuse or breaker which will carry that load but not much more. You always have to allow some extra because lamps and motors draw a starting surge. A few-Second surge does not heat the wire right away.
A "20A" breaker will pass 200A for 1 second, 100A for 10 seconds, 30A for 100 seconds, and 20A for over 1,000 seconds (~~15 minutes). It will blow instantly (0.025 seconds) at 300 Amps. (Other time-curves are available.)
By test or guess they pick a wire-size which will handle these currents "safely". The raw copper in open air can handle a lot of current, it is the rubber (now plastic) which is the limit, influenced by conductors per cable and conduit-fill and connector ratings.
This is usually conservative. Hot wires is wasted power. It is generally better to use a plenty big-enough wire. It wastes less, runs cooler. In the US, #12 wire can be rated 25A, but in general outlet use it is always fused at 20A. #12 wire can carry over 900A for some seconds. Likewise #14 will carry 20A steady at 60C, but must be fused at 15A.
So yes. If you plugged 13 Amps into a "10A" circuit, it will work, for a while. Possibly minutes.
They used to do this at work. One heater or tea-pot is 13 Amps. Two such loads is 26 Amps. On a 20A breaker, you can often get toasty toes and a pot of tea before the breaker blows. Or maybe not. They got good at running to the breaker box and re-setting for more abuse.
Constantly re-setting the breakers to hold an excessive load WILL melt the insulation and burn your house down. May take a while, but you really shouldn't.
There is a special problem in the US. Older wiring was mostly #14, which should be protected at 15A. It is too easy to change the breaker to 20A. Now instead of tripping the breaker, the wire runs hot for long periods of time. We often find old rubber-coat #14 toasted to death.