What's not been explained well here, is there's no one correct rule to apply. There are many different sets of rules which disagree on whether you ought to bid hearts or bid spades to show the support. (Although having found one major fit in hearts, you aren't bidding spades to find another one). The 'correct' bid depends on several 'variables':
- The system you are playing: Even 2/1 and SAYC apply different rules (let alone Acol!)
- Whether you adopt any Conventions (like Bergen, fit-jumps, etc)
- The length and strength of the suit(s).
(And that's without talking about partnership agreements which depart from underlying rules, interference, and majors vs minors).
Too many permutations to cover, although mostly they just differ by which fits are denied by a 1S bid, and whether the jump-raise in H promises 3 or 4 cards. Ignoring Bergen (and many others!) I believe it's common that:
In SAYC, responder raises with 3 hearts, (2H weak and 3H invitational) so 1S typically denies 3 hearts
In 2/1, responder wants 3 hearts for a minimum raise and 4 hearts for an invitational jump raise; So 3H might be misleading here and 1S followed by a 3H jump-raise is clearer.
EXCEPT! This responder's hand appears to have only 7 losers! so responder could insist on game themself, and stopping at 3H could be under bidding the hand!
Jacoby 2NT and Splinters generally need 4 card support; However I think 1S is generally a good and flexible course of action in most systems, hoping to show support with a 4H delayed-game-raise. But, when opener inevitably replies 2C! responder must either conservatively recognise wasted points in Clubs and Spades and settle for 3H - or see a cross-ruffing opportunity and optimistically plump for 4H. (Would you raise their 3H to 4H regardless with the undisclosed hand?) Checkback could be useful over a 1NT reply (although to investigate how well your hands match rather than looking for spade contract).
Generally in all systems, showing (major) support is always the first priority, however showing strength is an even Higher priority! - Thus, sometimes a delayed raise is the best description of support (it's not about any other suit, its whether the correct Strength and Trump Length are described under the partnership agreement).