Your parents taught you not to "waste water" because they paid the water bills. Thus the negative connotations associated with "wasting water" are purely (or at least overwhelmingly) economic. It does not logically follow that "wa$ting water" is bad for the environment. Your question "why is it bad..." is thus loaded. You've assumed something that shouldn't be assumed.
To break free of the cultural conditioning you were subjected to as a child, consider the following example:
A reservoir was established in a natural watershed in the hills near a small town. The elevation difference between the reservoir and the town is sufficient to provide an adequate amount of water pressure thanks to gravity alone. No pumps are needed. No electricity is consumed.
The catchment area for the reservoir is not subject to excessive contamination from windborne pollutants. It's not downwind of an industrial zone. It doesn't need much in the way of filtering, and certainly doesn't need treatment with poisons like chlorine.
The township is in a relatively dry climate (call it Mediterranean). So the water tables tend to be relatively low and surface vegetation doesn't have access to large amounts of water.
Finally, let's say that you live in this town, and you divert your greywater to a leach field in your back yard.
In this situation, clean water is made available to your house, and pours out of your taps, with no need for electricity or chemicals. When you wash the dishes or have a shower, the water goes into the leach field, which moistens the ground beneath your garden and lets you grow vegetables. Surplus water percolates harmlessly down through the soil and ends up back in the same unconfined aquifer that it originally came from.
The net effect of this is that your use of such a system hydrates the local soil and allows you to grow local crops — reducing the need to have diesel-fuelled trucks bring in vegetables from remote farms.
The more water you use, the more your soil gets hydrated, and the more produce you can grow.
If you "wa$te water" in such a situation (e.g. have long showers, use heaps of water in the kitchen), it is actually good for the environment — not bad. Since extra water lets you grow extra food, it's probably good for your wallet as well.
The "common knowledge" that it's "bad" to "wa$te water" is totally incorrect in such conditions.
Now compare and contrast that scenario with a city established in the middle of a desert, to which water needs to be pumped long distances from a power-hungry desalination plant on the coast, treated with chemicals in the process, and which (after use) is simply flushed down a common drain and ends up in some distant, polluted place — out of mind and out of sight of the residents. That is clearly bad for the environment and for wallets, so "wa$te is bad" would be true there.
Two different scenarios, two different outcomes.
"Wa$ting water" is almost totally an economic concept, not an environmental one. For it to be environmental you need to focus on how you get your water and what happens to it after the initial/primary use. Your original post doesn't go into detail on either of those.
If you expand your question to include details regarding where your water comes from, and what you do with it after you first use it, then we might be able to shed some light on whether you — in your particular situation — would be hurting the environment, helping the environment, or having no real affect on the environment, by using more water than absolutely necessary.
Whatever may be the case in your particular situation, that answer doesn't necessarily apply to your friend in the next town, or your next-door neighbour. Their circumstances may be different, so whether it's good/bad for you is irrelevant to them. Every property owner deserves to be assessed on their particular circumstances.
PS: Because an entire lifetime of cultural conditioning is hard (maybe even impossible) to break free of, it may help to stop using the phrase "wa$te water" entirely. Replacing it with "use more potable mains water than the absolute minimum required for a primary purpose, without regard for secondary uses" may be less convenient, but should get your brain thinking that there's much more going on that needs to be considered before jumping to conclusions.
Update...
Due to additional information being provided in the question and also in the comments of various answers, as well as some research.
Budapest is blessed with over 120 natural springs. These springs contribute a bit over a quarter of the city's total water consumption (especially to industry and public baths) and is the reason why locals believe they have abundant water.
Potable water is from wells dug into either islands on (or banks of) the Danube upstream. Sand is primarily responsible for filtering the water. UV and chlorine are responsible for disinfecting the water/pipes. Potable water is pumped into a number of elevated reservoirs. Electrical power for such power is provided primarily by Hungary's Pak nuclear plants. Overall, grid CO2 emissions have been steadily declining for the last two decades and the country is ahead of its renewable targets.
Wastewater is treated before being released back into the Danube. Chemical usage is proportional to waste content, not water content, so whether you concentrate the waste by using less water, or dilute it by using more water, makes no difference. Nutrient enrichment of the Danube is the same regardless.
So, in untangling the question and getting to an answer you first need to deal with the notion of "a water-rich country". The water used in public baths is not the same water that comes out of the taps. Because of the springs, Budapest has an abundance of non-potable water, but it does not have an abundance of potable water. Potable water is harvested and treated the same way that it is in countless other cities built on rivers — Budapest is not special in this regard. Budapest is not a potable water-rich city. Since you are talking about leaving the tap open longer, and only potable water comes out of the tap, that's an important distinction to make.
With that out of the way, the question of "how additional potable water use impacts on the environment" hinges on two factors: Electricity and chlorine.
Hungary's electricity can be characterised as "dominated by nuclear and steadily getting greener". In my view renewables > nuclear > fossil fuels (others are free to have their own opinion) so as far as I'm concerned, the environmental impact of additional electricity usage (for things like pumping and UV disinfection) are minimal.
Chlorine is primarily produced in the EU by electrolysis of salt water. The surrounding oceans and seas have plenty of salt water, and the EU has plenty of green energy, so recycling the chlorine from the water has minimal environmental effect as well.
In summary: Since you live in an apartment in a city, you are unable to close any meaningful nutrient loops so your waste simply enriches the nutrient content of the Danube downstream. Waste concentration is irrelevant. Your perception of abundance is not based on potable water, so you should put that to one side. The methods and energy used to create and distribute the potable water in Budapest are on "the good side of average". Chlorine — the only chemical that's noteworthy in this discussion — is abundant and easily harvested using energy from a green-ish grid.
Using additional potable water is definitely not going to make the environment better in your particular situation, but it is also not going to make the environment measurably worse. If I were you I wouldn't leave taps open and waste potable water just for the hell of it. Just use as much potable/tap water as you need. If using a bit more water (e.g. by having longer showers) makes your life measurably better, and you can afford the bills, then go ahead — no need to feel guilty. Take advantage of the public baths (and other uses for Budapest's abundant, non-potable spring water) whenever you can — it's a blessing that few other places enjoy.
tl;dr:
- Potable tap water: Use more only if it makes a noticeable positive
difference to your life.
- Non-potable spring water: Go nuts.
