There's a more detailed article on the subject here, written by the author cited in the OP, which may be the origin of the claim made in the Italian magazine:
The vegetarian dilemma
The elements of the argument, which justify the claim, are:
Most cattle slaughtered in Australia feed solely on pasture. This is usually rangelands, which constitute about 70% of the continent.
Each area of grain production in Australia has a mouse plague on average every four years, with 500-1000 mice per hectare. Poisoning
kills at least 80% of the mice.
At least 100 mice are killed per hectare per year (500/4 × 0.8) to grow grain. Average yields are about 1.4 tonnes of wheat/hectare; 13%
of the wheat is useable protein. Therefore, at least 55 sentient
animals die to produce 100kg of useable plant protein: 25 times more
than for the same amount of rangelands beef.
I've no reason to believe that these allegations are false: so I hesitate to call the claim "false" or "sensationalist".
However I have various reasons to believe that the claim is not widely/generally applicable as may be implied by the Italian magazine:
This FAO document says,
- Grazing systems. These are systems based almost exclusively on livestock production, with little or no integration with crops. They are mainly based on native grassland. In terms of total production, grazing systems are of lesser importance because they supply only 9 percent of global meat production. Of this, three-quarters comes from Central and South America and of the Organisation of Economic Cooperation and Development (OECD). Livestock interact in these systems with land, water and plant and animal biodiversity, especially wildlife.
- Mixed farming systems In mixed farming systems, crops and livestock production are integrated on the same farm. Globally, mixed farming systems produce the largest share of total meat (54 percent), and milk (90 percent).
- Industrial systems. These systems cover industrial types of production and small-scale urban or pert-urban production in developing countries. Both monogastric (pig and poultry) and ruminant production systems exist. They provide 37 percent of the total global meat production.
Therefore the argument is applicable only for 10% of meat (i.e. range-fed cattle and sheep).
Furthermore, almost all the relevent hits from the first three pages of the https://www.google.com/search?q=mouse+plague Google search are from Australia: so "mouse plagues affecting wheat" seems to be, more or less, a specially-Australian problem.
Thirdly, the author is IMO cherry-picking his foodstuffs:
Not all meat is cattle (pigs, chickens, etc. are monogastric) and don't eat grass.
Wheat isn't the only food for human vegetarians (but I suspect that wheat is especially susceptible to mouse plagues), so by picking "wheat" he is picking a 'worst case' crop for mouse-killing.
Nevertheless, articles such as Criticisms of Environmental vegetarianism may be right in saying than being vegetarian is insufficient:
According to Cornell scientists, "the heavy dependence on fossil
energy suggests that the US food system, whether meat-based or
plant-based, is not sustainable." but they also mention that:
"lactoovovegetarian diet is more sustainable than the average American
meat-based diet. " 
Some environmental activists claim that adopting a vegetarian diet may
be a way of focusing on personal actions and righteous gestures rather
than systemic change. Dave Riley, an Australian environmentalist,
states that "being meatless and guiltless seems seductively simple
while environmental destruction rages around us," noting that animals
can contribute to the food chain.
Bill Mollison has inconsistently argued in his Permaculture Design
Course that vegetarianism exacerbates soil erosion. This is because
removing a plant from a field removes all the nutrients it obtained
from the soil, while removing an animal leaves the field intact. On US
farmland, much less soil erosion is associated with pastureland used
for livestock grazing than with land used for production of crops.
Robert Hart has also developed forest gardening, which has since been
adopted as a common permaculture design element, as a sustainable
plant-based food production system.
Eating range-fed Australian cattle is not feasible for the majority of the people (vegetarians or otherwise) in the world, and for that reason the article isn't wholly or generally true (because it's impossible).
If you are interested in this subject (i.e. if you want to do something about it) there may be other options which you can explore locally (in whichever country you are living in): for example, "organic" farming might be less deleterious to wild-life ... but deciding that would be a topic for a different question.
I am inclined to praise the author for at least making me think more about the subject.