You're thinking of it like "cumulative" + "lifetime incidence"; they're thinking of it as "cumulative incidence" and answering the question "cumulative incidence over what time period?" as "over the lifetime". "Lifetime cumulative incidence" might be a better way to write it from a statistical terminology perspective but certainly sounds more clunky to me; rules for adjective order are imprecise, inconsistent, and not typically taught (but learned implicitly nonetheless), but you might get a better answer on that from a linguist.
"Cumulative incidence" is used to refer to the incidence over an entire time period (where you have to specify the time period to be meaningful). You could imagine an "instantaneous incidence" instead that would have different units and shape, as it would be a time-varying function. Usually that's just called "incidence" or "incidence rate" and is expressed as "events/time" rather than "events/person". Cumulative incidence is the integral of incidence over some time window.
This happens to be the first page that came up when I searched looking for something explaining this and I think it covers it fairly well.
If you follow the citation chain from the original article (which directly quotes a statement from the referenced paper that is completely from another paper) you get to:
Kujala, U. M., Sarna, S., & Kaprio, J. (2005). Cumulative incidence of achilles tendon rupture and tendinopathy in male former elite athletes. Clinical Journal of Sport Medicine, 15(3), 133-135.
In this paper, they include two measures of cumulative incidence: cumulative incidence up to age 45, and cumulative incidence up to the age of study ("former elite athletes" median [range] 69 [54-97]). They call the latter "lifetime", though these people are at least alive enough to answer a survey so the meaning of lifetime here is in "years already lived by the study population" rather than an "average lifespan" or something like that. These authors also use "lifetime cumulative incidence" rather than "cumulative lifetime".
I do see a lot of papers that use "lifetime incidence" to mean "cumulative incidence" where the time period taken is a "lifetime" so you could certainly argue that it's redundant to include both, but that's usually true for cumulative incidence over any time period, in that if you express an incidence as a proportion of a population over time you're clearly talking about a cumulative incidence. The extra word just makes it a little more clear; most language includes some redundancy.