I assume that by "channel" you mean "group"
On any platform, you can use the lens icon to search for messages. Once you tap/click on it, an user silhouette π€ icon will appear. It allows to filter messages from a specific user, yourself included
If by "channel" you mean "channel", then there's no way to know which of the posts in the channel were sent by yourself, since they all appear as they were sent by the channel
If you want to see a list of all messages you've sent across all chats, then there's no way to do that unless you script something using the users API
Yes, they are two different apps actively developed by two different people (A source code, K source code). They are the two winners of a contest Telegram held a few years ago, but they haven't' decided which one will be the default/standard webapp yet, so you can use both. I think web.telegram.org still picks a random one for you the first time you access it