First you don't need to repost your message within 20 minutes of your first. This isn't a chat room so you won't always get a response within minutes.
Simply put Unlimited is a format in which any card is allowed except Ancient Mew and Birthday Pikachu which are just collectors cards really. Now what most tournaments including city championships are is a format called Modified. Modified has been different year to year but it is always the last few sets with most of the older cards restricted since newer players would have trouble accessing them. The current modified format is any cards from
Expedition
Aquapolis
Skyridge
Ex: Ruby & Sapphire
Ex: Sandstorm
Ex: Dragon
Any Nintendo Promo cards. Starting with Kyogre and Groudon and going up.
Also any cards that were reprinted in these sets such as Potion are allowed even if that copy wasn't in those sets. Like you can use a base set Potion in your deck because it was reprinted in Expedition and Ruby and Sapphire. Most people post these kinds of decks because all of the big tournaments only allow these cards. I don't know what you mean by the other kind but people sometimes refer to the old modified formats as different things.