"Proton pump" is a broad category of proteins rather than a specific pump.
The drugs called "proton pump inhibitors (PPIs)" to reduce stomach acid target a specific proton pump, the hydrogen/potassium ATPase.
Of course, it is possible for drugs to have off-target effects at other proteins, especially similar ones. It is also possible for side-effects of PPIs that are secondary to the reduction in gastric acid, which can affect digestion and absorption of certain nutrients.
Lysosomes and other organelles use a different proton pump, the V-ATPase. If you search for papers involving specific PPIs and the V-ATPase it does appear that V-ATPase can potentially be affected by PPIs, and this has been considered as an anti-tumor strategy by some:
Fais, S., De Milito, A., You, H., & Qin, W. (2007). Targeting vacuolar H+-ATPases as a new strategy against cancer. Cancer research, 67(22), 10627-10630.
I'm guessing the article you linked to is referring to this paper:
Yepuri, G., Sukhovershin, R., Nazari-Shafti, T. Z., Petrascheck, M., Ghebre, Y. T., & Cooke, J. P. (2016). Proton pump inhibitors accelerate endothelial senescence. Circulation research, 118(12), e36-e42.
In this paper, cultured endothelial cells are incubated with the PPI esomeprazole at 5 or 10 μmol/L. They found increased pH (reduced acidity) in intracellular comparments imaged with a pH-sensitive dye, and various symptoms of endothelial dysfunction after esomeprazole treatment.