As stated in the question, there seems to be only three distinct times that the spell duration can end (if not specifically stated in the spell). That would be the beginning of the turn, the end of the turn or sometime between. The duration ending any other time would not fulfill the 1 round duration time limit.
Here I will make an assumptions. The makers/writers of the game seem to go with a more simplistic approach. Keeping track of what the order of actions was from round to round seems rather complicated and dull. I mean if you do action, bonus action, movement one round and in the next you do movement, bonus action, action in the next, your one round spell would have expired before you take your action. So I am going to assume that the makers/writers did not intend for people to keep track of the order of their action from one round to the next. That takes the spell duration ending “sometime between” option out and just leaves “at the beginning” or “at the end” of your turn.
So let us take a look at the spells that have a duration of 1 round. I will only be looking at spells from the Players Handbook (PHB) and the Elemental Evil Players Companion (EE). I think this will be enough of a test pool to draw a conclusion from. To save space I will only paraphrase the pertinent part, but will give book and page number of where the spell can be found.
First let us look at spells that specify when they end in the description:
Command (PHB p223) = Target will follow command on its next turn
Shield (PHB p275) = beginning of your next turn
Chill Touch (PHB p221) = start of your next turn but also has Until the end of your next turn
Guiding Bolt (PHB p248) = Until the end of your next turn
Blade Ward (PHB p218-219) = Until the end of your next turn
Teleportation Circle (PHB p282) = until the end of your next turn
Absorb Elements (EE p15)= Absorb till beginning of turn, damage till first hit on next turn
So Command, Shield and Chill Touch (when cast on a living creature), by their very description will not last a full round even though their duration is one round. Chill Touch (cast on undead), Guiding Bolt, Blade Ward, Teleportation Circle and Absorb Elements can or will be longer than the 1 round duration. This sets the precedence that a one round duration can mean a little less or a little more than one round.
So now let us take a look at spells that don’t specify exactly when they end.
True Strike (PHB p284) = On your next turn, you gain advantage on your first attack roll against the target, provided that this spell hasn’t ended.
Color Spray (PHB p222-223 = until the spell ends
Message (PHB p259) = no specified time
Sending (PHB p274) = no specified time
Transport Via Plants (PHB p283) = for the duration
True Strike has a number of questions on how it works. While the spell does not say it explicitly, but does say you get a benefit on your next turn. The consensus I see is that it lasts until the end of the caster’s next turn. Since if the last action you take is a bonus action to attack you would get the advantage as long the caster had maintained concentration.
Transport Via Plants is close enough to Teleportation Circle in effect that I feel comfortable saying it also should last until the end of the caster’s next turn. Then you can have the image of the caster waving everyone through and then stepping through himself rather than him running in first.
The rest of the spells do not specifically say they last less than a round and so they should last into the caster’s next turn. This is where my assumption comes into play. that the duration of a spell will not arbitrarily end in the middle of the caster’s round. So by default they will last until the end of the caster’s round.
Unlike abilities like rage where the user gets the benefit immediately, the caster doesn't start getting the benefit until after the end of their turn. If the spell were to end at the beginning of the caster’s turn, then the caster would not benefit fully from the spell he cast (especially since it could have been the last action he took in the round). His allies would get the benefit, but the caster would not.
So how long does a 1 round spell last, unless annotated in the spell description that it ends earlier, a 1 round spell lasts until the caster’s next turn. At least that is my conclusion.
I feel I am still learning the game so let me know if I have made an incorrect assumption or went off on my logic. Thanks.