When discussing the First World War, there is confusing terminology when it comes to battles. Battles can be as general as the German Spring Offensive (literally translated as Kaiser's Battle from German), or as specific as the relatively-small action at Cantigny.

Even in the smallest skirmishes of the war, as you eluded to in your question, there were multiple phases to combat, as necessitated by the fact that each "trench" was in reality a deep system of crisscrossing or polygonal trenches interconnecting with one another, forming several lines to prevent an enemy breakthrough. When one abstracts this even further, there are multiple lines of these so-called "trenches," as made apparent by the fact that the war in the Western Front didn't end until 1918 despite countless breakthroughs before then.

A misconception that you seem to have is that the attacking side would give up after an assault became too costly. In any battle large or abstract enough to be named, this would never have occurred. Offensives took a lot of time, resources, risk, and care to prepare for, and thus commanders would rarely squander their opportunity despite mounting casualties. In all offensive side would always gain some amount of territory before deciding to dig in and defend their newly-claimed ground.

Contrary to what popular media portrays, soldiers of the First World War did not attack in massive, easy to target groups. While this type of strategy may have been slightly applicable to the early-war stage, especially that of the French in 1915, but still not to the degree you'd be expected to believe. Instead, staggered waves of more spread-out troops were used in order to create and exploit gaps in enemy lines and well as build momentum, while at the same time minimizing casualties from destructive weapons such as machine guns or artillery. Sophisticated systems of multiple squads were indeed utilized, but didn't march all together in a single mass only to be cut apart.

Combining these ideas, we can begin to see the full picture through which tactical doctrines were produced. It was always a given that the first line of the trench would be taken, as no attacker would quit the battle in the face of most conceivable casualties. Strategy therefore relied on defense-in-depth, using reserves safely placed in rearward lines to retake the ground lost by the initial attack. In this way, battles often saw a tug-of-war between the two sides, where each would attempt to oust the other from their newly-acquired positions in succeeding waves of men. The goal of the attacker would be to continue making use of their momentum while the goal of the defender was to break it, and possibly exploit the aggressors over-stretching or overworking themselves. Another factor leading to the inevitable gain of land from an offensive was that the attacking side, having spent considerable pain to prepare for said offensive, would usually have the other side both outmanned and outgunned, offsetting the strength of defensive fortifications.

So, what makes a British offensive specifically a "British" one? Well, as you can probably tell by now, that identification would result from which side spent the resources to prepare for the start of the assault. Additionally, which side eventually gained the most ground would virtually always correlate with which side started the attack. The challenge of contemporary commanders wasn't whether or not they could gain ground, but rather whether or not they could make doing so cost-effective.