I thought "responsibility for" is about the responsibility for an action or an object, and "responsibility to" is about the responsibility with a person, like if I say "I must have the responsibility to her for losing her money", it means that because I lost her money so I must have the responsibility with her. But when I searched on the web, I found a page titled that "Responsibility to our environment", environment is an object and why they use "responsibility to" instead of "responsibility for"? Is that usage of that page correct? Thanks for your help!
"Responsibility for" would precede a noun or a noun phrase. Example:
A parent has a responsibility for their child. A student has a responsibility for doing their homework.
Here it means that the object is under the care or supervision or purview of the subject.
"Responsibility to" would precede a verb. Example:
A parent has a responsibility to teach their child. A student has a responsibility to do their homework.
Here it means that the verb is expected to be handled/enacted by the subject.
In the example you provided ("Responsibility to our environment"), it's a less specific form, and isn't
[responsibility to] [our environment]
[responsibility] [to our environment]
meaning that everyone has a general responsibility, and that responsibility is directed toward our environment -- that everyone should act as though they are responsible FOR the environment. It is a much more situational usage that I would recommend avoiding, as to me it seems somewhat strange English.
The OP’s assumptions about how these two prepositions work after responsibility and responsible are generally correct. If one is responsible for X, then X is either the action that would constitute discharging the responsibility, or the result of that action, or some person or thing that this action is centered on. If one is responsible to Y, then Y is some person who would be entitled to complain and demand some sort of remedy, if the responsibility is not discharged. For example, if one promises to Y to take care of X, the promise creates a responsibility for X, to Y. A babysitter is thus responsible to the parents, for the baby’s wellbeing and safety.
Now, how does that work with respect to the environment? Everybody who is concerned about the environment agrees that human beings have a responsibility for the environment, that is for its flourishing, for ensuring that it is not damaged, etc. Behind that apparent disagreement, there is, however, a great deal of disagreement about what ethical considerations underlie the environmental concerns. Some people, when asked why they are concerned about the environment, might respond that the damage to the environment can affect the availability of crops that humans use for food, or cause flooding that would disrupt human life in coastal regions, and so forth. These people believe that our responsibility for the environment is a responsibility to the humans (including future generations of humans) who may be disadvantaged by the damage to the environment; their concerns about the environment are ultimately anthropocentric.
There are, however, also people who regard the natural environment as valuable in itself, independently of any impact on humans. They think of the environment (or of particular ecosystems) as something that demands respect because of what it is. The proponents of this approach to the environment reject the idea that the responsibility for the environment is a responsibility that we have to other humans; they think of it as a responsibility to the environment itself. The author of the text that caused OP's puzzlement probably chose the preposition to, rather than for, to express an allegiance to this way of thinking about the environment, and to make it clear that the text is not based on an anthropocentric approach to it.
Purely based on what sounds natural to me as a native speaker, I would say you are responsible for someone or something, and you have a responsibility to someone or something.
In other words the difference is responsible versus responsibility. But there is also a slight difference in semantics I think.