Is there a definition for pedophile who has acted on urges and one who has not?

I have being looking for a way to differentiate between someone who is sexually attracted to children but has not acted on the urge and a pedophile who has actually acted on their urges.

Pedophile is sexual attracted to children. Pederast is a man who indulges in sex with boys.

Is it sexual predator? or does it change to being a pedophile rapist? Since children are unable to consent it would be rape.

It's come up lately in a lot of videos I watch where they have shown pedophiles coming forward and talking about their lives. Many of these seem to have the urges but have not crossed the line of sex or interacting with children. However we call both types pedophiles.

There is no single word that I know of. I think RaceYouAnytime's suggestion of "child molester" fits well.

Other appropriate phrases for someone who has sex with children are "child abuser", "sexual abuser", "perpetrator of child sexual abuse". The three-word phrase "child sexual abuser" is formally ambiguous about which party is the child, so it may not be the best choice.

The use of "pedophile" that you mention is not really appropriate for a formal context due to the vagueness that you mention. The words p(a)edophile/p(a)edophilia have a specific definition in psychology that is not equivalent to "child molester/child molestation". As Wikipedia indicates, in psychology "pedophilia" is defined as sexual attraction (often exclusively) to prepubescent children. Child sexual abuse on the other hand covers sexual activity with children regardless of how the perpetrator felt about it; and in a legal sense, it encompasses abuse of adolescents who are still below the age of consent in their jurisdiction.

This may be a better question for a forum of psychiatrists and psychologists. I think your question goes beyond a linguistic one. That said, here's my attempt at an informed answer that has an emphasis on terms used by researchers and clinicians in the mental health field.

A quick search on Google Scholar yielded a term that may be useful to you. You might use the term non-offending pedophile for someone who is sexually attracted to children, but does not molest children. A journal article with that term as a title can be found in Current Sexual Health Reports, September 2016, Volume 8, Issue 3, pp 121-128. You will note that the definition of non-offending includes no molestation of children and no use of illegal media that exploit children.

Offending pedophile would then be the term for a person with sexual attraction for children who does sexually molest children. Colloquially, child molester is a useful term, but it probably does not include pedophiles who use child pornography. Predatory pedophile is a term I came across in my searches and it seems useful as well, as suggested in a previous answer.

There has been a lot of controversy about the definition and categorization of pedophilia in the "Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition" colloquially called the DSM-5, the latest edition of the diagnostic manual of the American Psychiatric Association published in 2013. You can read a good summary here about what the DSM-5 says about pedophilia. Pedophilia is considered a paraphilic disorder.

In DSM-5 the term paraphilia is defined as “any intense and persistent sexual interest other than sexual interest in genital stimulation or preparatory fondling with phenotypically normal, physiologically mature, consenting human partners.” Paraphilias, however, may not necessarily classify as “intense and persistent” but rather preferential sexual interests or sexual interests that are greater than nonparaphilic sexual interests. The addition of the word “disorder” to the classification of paraphilias is new to DSM-5. (http://www.psychiatrictimes.com/dsm-5-0/dsm-5-and-paraphilias-what-psychiatrists-need-know)

Pedophilic Disorder is a DSM-5 (Diagnostic and Statistical Manual of Mental Disorders, fifth edition), diagnosis assigned to adults ( defined as age 16 and up) who have sexual desire for prepubescent children (American Psychiatric Association, 2013a).Any behavioral expression of Pedophilic Disorder is a criminal offense in the United States, Canada, and Europe, as well as most other places in the world. Some authors differentiate between having deviant desires for children which are ego-dystonic and resisted, causing guilt, shame, and distress, vs. desires which are indulged through fantasy, associating with other pedophiles, possession and trading of pornographic images, or direct observation, self-exposure, or physical contact with a victim (Harvard University, 2010; Vachss, 2013)

If you can wade through the psych jargon, this scholarly article (Arch Sex Behav (2015) 44:1127–1138) gives a good overview of how the change in sexual norms in culture influences mental health and legal ideas of what is pathological or criminal. The move is away from the idea of deviance and toward the idea of consent between adult partners and lack of harm to self and other.

The only designation I am familiar with would be for one who does seek out children for sex and they would be a predatory pedophile.

Predator- A person who ruthlessly exploits others.