If the police or other authorities of Sri Lanka do not have witnesses or other evidences supporting that this American committed the crime, there is little that can be done. The police of Sri Lanka could ask the police of whatever locality the person is now in to interview the accused, which they might do as a favor, but are not obliged to do. If they do such an interview, the content would probably be sent to the Sri Lankan authorities. It might or might not be of help to them.
If the Sri Lankan authorities can assemble some evidence, they could apply for extradition. That is a matter which depends on the existence of a treaty. According to 18 U.S. Code § 3181 there is an extradition treaty between the US and Sri Lanka, which became effective on Jan. 12, 2001.
Extradition requests are normally subject to review by the courts and diplomatic authorities of the sending nation, that is the country that is requested to extradite a person. The request may be refused if there is not sufficient evidence, or if the charge is not a crime under the laws of the sending nation, or for other reasons specified in the law of the sending nation.
In the hypothetical case in the question, the main issue would probable be whether there was sufficient evidence. If it were claimed that the accused could not get a fair trial in Sri Lanka, or would be likely to be subject to torture there, that could also be a reason for refusal.
In the US this is handled under 18 USC 3184 This provides in relevant part that:
If, on such hearing, [the judge or magistrate] deems the evidence sufficient to sustain the charge under the provisions of the proper treaty or convention, or under section 3181(b), he shall certify the same, together with a copy of all the testimony taken before him, to the Secretary of State, that a warrant may issue upon the requisition of the proper authorities of such foreign government, for the surrender of such person, according to the stipulations of the treaty or convention; and he shall issue his warrant for the commitment of the person so charged to the proper jail, there to remain until such surrender shall be made.
I do not know the details of criminal procedure in Sri Lanka, so I don't know what, if anything, a court might do about such a case. In the US, a court would have nothing to do in such a matter until and unless the prosecutor was willing to present formal charges. Such charges would have to establish that there was probable cause to believe that a crime had been committed, and that the accused had comitted it, before a trial could be held.
The Sri Lankan authorities could also request that a relevant US prosecutor file charges and try the accused in a US court. In some cases the US will try US nationals for offenses committed outside the US. But in the kind of case described in the question, this would not be usual.
If Sri Lankan law permits, the authorities there could also try the accused in absentia. Many countries and many people regard this as fundamentally unfair, but it is sometimes done. In such a case, the person convicted in absentia could be arrested if s/he ever came within the jurisdiction of Sri Lanka. Aside from that, Sri Lanka cannot bring the person to court unless they apply for and get extradition, or the person travels to Sri Lanka again and is arrested there.
As to the specific questions:
if a good lawyer was stating the question above, what would their precise terms be?
There are not very many specific terms to be used. The person said to have comitted the murder (or other crime) would probably be refereed to as "the accused" or possibly "the alleged criminal". If there is a specific term for the particular crime in Sri Lankan law, it would be used. If it were a real case, details such as the time and place of the crime would be given.
will this whole hypothetical situation fall under the case study of criminal law? I'm thinking international laws as well, but then I don't know for sure what international laws are, so there's that.
Yes, this is an aspect of criminal law, albeit one that most lawyers in criminal law practice never have occasion to deal with. In a sense there is no such thing as "international law", that is, law binding on countries. What is known as "international law" is the complex of treaties and agreements between various countries and the way in which they are given effect. Nations never have to enter into treaties, although there can be great pressure to do so. A nation can always denounce a treaty, although there may be consequences such as political or economic pressure, or in extreme cases, war.
"international law" is also the accumulated precedents for how nations have interacted in the past over legal issues, but this is only binding in future cases to the extent that a given country's laws or courts treat it as binding, or that a current treaty makes it so.