Really good question, and not one that I could immediately find a definitive answer to. Everything seems to hinge on whether "build" implies all the same things as "put into play".
In the rulebook to the original game, the term "build" is used many, times, often as part of the phrase "build for free" (which proves that building can happen in ways other than the conventional way).
The phrase "put into play" only appears once:
Each Age has 6 game turns. During each turn the players
simultaneously put into play one card.
Which suggests that building always counts as putting into play, but doesn't prove that the reverse is also true.
Sometimes these things are just the result of sloppy phrasing in one language or another, but I looking at a French version of the Leaders rules and the wording is exactly the same: Solomon "puts into play", not "builds".
In the absence of official errata or clarification I would have to say that I don't think Solomon should get 2 coins from Xenophon, although I do think Halicarnassus should. "Building" and "putting into play" are sufficiently careful and distinct terms that it's possible to imagine the game designers might not have meant them to be synonymous. Why on earth this should be is of course another question entirely. Maybe they feared that games could end up being seriously dominated by the OBVIOUSLY BROKEN Solomon/Xenophon combo. </tongue in cheek>
ETA: Having spoken to the game designer Antoine Bauza on Twitter about this I'm going to change my answer and say that putting a card building card into play is exactly the same thing as building it, as this seems to have been the intention. It feels like the choice of wording was to allow the possibility of putting other types of cards that are not buildings into play with Solomon... Anyway, I'm just trying to double-check the intention behind the wording with Antoine, but now that we are no longer "in the absence of official clarification" I'm going to say that sitalnax's instincts were right, putting a building into play is to all intents and purposes the same thing as building it!