But if we are going to go by definition here, pedophilia is characterized by any 16 year old or older who is attracted to prepubescent kids. That means that if a 16 year old boy is attracted to a 13-year old girl, he is a pedophile. To be honest, this definition doesn't make sense because girls these days look much older than they really are and considering the age gap between them, it's only 3 years.
You could argue that you're talking about a 50 year old man wanting to marry a 9 year old girl, but it's not really that different because honestly guys are attracted to females point blank. If the girl is attractive, a guy will be attracted to her regardless of her age. There are 70 year olds who are married to 18 year olds, that's a 52 year difference. There are 13 year olds married to 14 year olds; both being "underage" and pubescent. If you want to follow a western ideology, then when it comes to marriage, age shouldn't matter, right??
Marriage should not be something that strips away from a person's life, it's something that is a blessing. Anything that makes it feel otherwise is poisonous. It's not something that should happen in secret or as a business transaction (like some cultures make it where they "sell off" their daughters). It's a celebration of two people who believe they are compatible for the rest of their lives, who will balance one another in faith, and who are following the laws ordained by Allah subhanahu wa ta'ala. That being said, the majority of scholars believe that is preferable for girls to marry once they reach the age of puberty (sexual maturity), and not before. But for the rare cases that it does happen, then it depends on the interests of the girl first and foremost. Girls these days are sexually curious and having intercourse at 9 years old. I personally think marriage is much more suitable and safer for them than in a culture that condemns "child marriages" but does very little in preventing a sexually geared atmosphere where they can easily be raped, molested, and pregnant. So there's really nothing that says that it's unsafe for girls to get married or be in a relationship at that age unless the marriage is unhealthy and destructive in nature which is unislamic. And when it comes to pregnancies, there are complications in every pregnancy whether the woman is an adult or a child. Some women die too..so really that's not a strong argument. The child marriages you imagine are the ones that come out of poverty, backward cultures, and ignorance. The pedophiles you think about are the ones where men can be attracted to boys just as well as girls, who are a danger to children, and who victimize them. None of this is allowed in Islam.
Now with your comment regarding why shouldn't all men emulate the prophet pbuh and marry children? Because the prophet did not marry for the sake of his desires and he married a range of different women and he treated them all very well and they loved him to death. Marriage in general is sunnah. It's not obligatory. Marrying children isn't obligatory either. It's just something that is allowed if something of that case happened where a man finds himself interested in a girl and she is of a certain age. In the US there are many states which have no age minimum for marriage. Isn't that crazy? But the difference with the westernized world and Islam is that Islam has universal rules. It's not based on the whims and desires of people where it changes ever so often. The connotation of pedophilia often comes with a lot of perversions that are not allowed in Islam, and there is no exception to it whether the man who wants to marry is 13 or 70.