The media throws around the term "immortality" when talking about efforts to extend healthy life, with little concern for the dictionary definition. Advocates for radical life extension have in the past used physical immortality as a alternative term for the concept of agelessness, in which aging is controlled but all other causes of death still exist - which is another change of meaning. Some people find this a distraction, an annoyance, something that makes it harder to conduct advocacy and fundraising for current and prospective longevity science. Convincing the world that rejuvenation therapies are a viable near term prospect, given sufficient funding, is challenge enough without the peanut gallery.
It isn't clear whether or not dictionary definition immortality is possible in this universe, and if it was the entities enjoying it would be very different from the present human model of existence. Even scaling up to a reliable life expectancy of a million years would require considerable technology-assisted change and expansion. Such long-lived beings would probably be something akin to distributed collections of hardened, space-faring, automated computational factories. In that sense, we stand a long way removed from even the lesser challenges of living for a very, very long time. The problems of today, in which we take the first steps towards treating aging as a medical condition, so as to add the first few additional decades of healthy life, are those of the first rung on an extremely long ladder - and they are hard problems. If we don't focus on them, there is every chance of failure to progress soon enough to matter for most of us.
It's not uncommon, especially for outsiders of a given field, to use an inappropriate word to indicate a more complex concept than the word itself conveys - maybe because they think that the two are close enough or possibly because they just don't see the difference. For this reason, it's likely that each field has its own unspeakably profane word; in the field of rejuvenation, that word is the dreaded I-word: immortality.
Whether or not immortality is possible is an intriguing question, but it is decidedly off-topic in the field of rejuvenation, because rejuvenation is not immortality. If a universal antiviral drug existed, able to wipe the floor with every conceivable virus, you wouldn't call it an immortality drug, because right after leaving the doctor's office where you got your miracle shot, a grand piano might happen to crush you after a 50-story free fall, and the antiviral drug wouldn't be especially effective against that particular cause of death. Similarly, rejuvenation would save you from death by age-related diseases, but again not by falling grand pianos.
Yet, both people and the media keep talking about "curing death" and "immortality pills" when the actual topic is rejuvenation biotechnology; this is a cause of particular annoyance to Dr. Aubrey de Grey, whose pioneering work is constantly called an "immortality quest" and similar things. Since immortality reasonably seems a pipe dream, this results in a gross misrepresentation of the entire field and a lot of unwarranted bashing of completely legitimate medical research whose only fault is that it aims to prevent the diseases of aging rather than just coping with them.
The same story is true of negligible senescence. If a successful rejuvenation platform were implemented, people would still age biologically, but we would have therapies capable of undoing such aging. Through periodic reapplication of these therapies, the hallmarks of aging would always be kept well below the pathology threshold. In other words, we would still senesce (that is, age), but our level of senescence would stay negligible - that's where the term comes from. Yet, many people keep calling negligible senescence immortality just like they do rejuvenation biotechnology, whether deliberately or by genuine mistake, thereby providing an excellent strawman for needy critics to beat.
Negligible senescence is the expected result of truly comprehensive rejuvenation biotechnologies, and yes, if we got there, our healthspan would be vastly increased, and consequently, so would our lifespan; if you were in perfect health for longer than, say, 100 years, it is a disarmingly trivial consequence that you would live for longer than 100 years. However, whether a negligibly senescent person then lives on forever or not, or ten thousand years from now, someone beats the odds and comes up with a fancy immortality switch, is an entirely different matter that is beyond the scope of the field of rejuvenation biotechnology.