So if I'm using a "baseball bat" two-handed in your world it would be a simple blunt weapon (versatile).... I'm assuming a d8 (like a quarterstaff)?
Pretty much, although the way we'd think about it would be mechanics first and then naming it second. So if someone was proficient in Simple weapons or quarterstaves, they could select a 1d6 Simple bludgeoning weapon (versatile 1d8) and call it a 'baseball bat'. Or a 'greatclub'. Or a 'tree branch'. Or whatever. And we actually don't even distinguish between the three weapon types either (since for instance the weapons chart does currently list any piercing or slashing 1d6 (versatile 1d8) weapons.
Simple melee weapons are normally 1d6. If they are also Light, Thrown, or Finesse they are 1d4. If they are Versatile they can go up to 1d8 but can't have any of the other three properties. I don't bother with the 'greatclub' category of strictly a 1d8 two-handed weapon because it's pointless.
Martial melee weapons are normally 1d8, Finesse *or* Versatile (1d10). If they are Light they are 1d6. If they are two-handed they are 1d12 regularly, 1d10 if they have Reach. (As a special dispensation, if a player wants to do two-weapon fighting, I will allow them to fight with weapons that are 1d8 main / 1d4 off, rather than 1d6/1d6 even though the main hand weapon technically isn't Light. But as I like the fencing sword / main gauche combo aesthetically, I don't worry about it.
Once the player has chosen what category of weapon they are going with... they can fluff it out however they would like and then I'll assign what category (bludgeoning, piercing, slashing) it applies to.