Because Americans in Timbuktu pay federal income tax, unlike those in Puerto Rico, and there's "no taxation without representation." D.C. residents get to vote for president for the same reason.

The US doesn't have a presidential election. Get that idea out of your head. Instead, all 51 states^* have their own presidential elections on the same day.

This selects a set number of electors for each state. Those electors get together weeks later, and they decide who is President.

To be eligible to vote, you have to be a US citizen resident in one of those states. Most (all?) states have mechanisms for allowing residents who have to be out of town during the election to vote (eg: in Timbuktoo or even in Puerto Rico). However, you have to have a state to send that ballot to.

Puerto Rico, like other US territories such as Guam and American Samoa, is not a state. Therefore, they (and people resident there) are left out of that system. However, as full-blown US Citizens, they are allowed to move to any US state of their choosing and vote there.

This is fundamental to the design of the US Constitution, and can only be changed through the Amendment process. However, how a state picks its electors is up to each state. A state could, out of the goodness of its heart, "adopt" a territory and let its residents vote in that state's election. However, unless something like the National Popular Vote Interstate Compact is adopted, that seems quite unlikely, as it would dilute the power of the state's own voters for that state's set number of electors. However, if the compact goes into effect, there'd be no real downside to doing that, so we could then see US citizens resident in territories being allowed to vote for President.

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^{* - Technically there are only 50 states, but residents of the stateless federal district of DC are allowed to vote there as well, via a special constitutional amendment that was passed in 1960.}

"What is the legal reason why an American visiting Timbuktu can vote for president but an American resident of San Juan cannot?"

An American who is a resident of San Juan, and who is visiting Timbuktu, can not vote for president.

The US has a concept of "states" and a concept of "territories". There are various differences, particularly in relation to taxes and voting.

The answer to the question "what is the 'legal reason' that residents of territories do not vote for president?" is simply "because it is the law", a historical issue.

The issue of the exact details of what determines "residency", in the US, in various territories/states, is a detailed issue often seen discussed in (say) a tax discussion.

The issue that Americans can (or can't) vote while visiting overseas locations such as Timbuktu is completely fatuous and irrelevant.

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Regarding the curiosity "which state (/territory, /city, /county etc) does a US citizen vote as if in, when, overseas for many years", this is the common issue and it's simply (what else could it be?) your previous address.

Here's a detailed and clear explanation of the details about US'ers who live overseas for a long time:

Short article

Residents of Puerto Rico are not entitled to vote because Puerto Rico is part of the United States commonwealth. A territory part of the US Common wealth is "An organized United States insular area, which has established with the Federal Government, a more highly developed relationship, usually embodied in a written mutual agreement", according to the Department of Interior. Moreover, an "insular area" is "A jurisdiction that is neither a part of one of the several States nor a Federal district" (from that same DOI link).

Hence, because it isn't a state nor a federal district, it does not have any voting rights in Congress, nor does it have electoral college votes. The exception to this rule is the District of Colombia which was awarded electoral college votes via the 23rd Amendment to the US Constitution, in 1961.

US citizens who live abroad are entitled an absentee ballot as stated on the website of the Department of State. More specifically, "U.S. citizens living outside of the U.S. register and vote in the state and county where they last established residence (domicile) in the U.S. before moving outside of the country", cf. here. As such, a US Citizen living outside of the US whose previous residence was in a US State or in a Federal District is entitled to vote when they live abroad.