Marek, I have worked today. The present perfect is used to describe actions that happened at an indefinite time in the past up to and including the present moment. This sentence means that you worked sometime during the day today, you may STILL be working now. The tense is created by using the PRESENT TENSE AUXILIARY HAVE and the past participle of the verb 'work', so the present perfect always has a relation to the present. I have worked today. So Saturday is not (or has not been) a free day for me. I had to work today. "Had to" expresses past necessity. The sentence means that you were obligated to work today. You are no longer working at the present moment. I had to work today. So....Saturday wasn't a free day for me. Everything is in the past. You are finished working and so Saturday was not a free day. You could also say. I worked today. So Saturday wasn't a free day for me.

In your first sentence you have used a present perfect. You can use it for past experiences when you dont say exactly when it happened, recent past actions and with yet and already for emphasis. Use it for unfinished states (non-action verbs) which star in the past and are still true now. Examples: I have been to London but I have not been to Oxford (past experiences) He has just arrived at the airport (recent past action) I have already done my homework. Present perfect or past simple? Let's see. I have been to London twice (in my life up to now) I went to London in 1995 and 2007 (on two specific occasions) How long have you been married to Susan? (You are married to Susan now) How long were you married to Mike? (You are not married to Mike now) In the second sentence the native speaker use a modal verb "Have to" is used to express an obligation.