Learn English – “get used to cycle” or “get used to cycling”

I would say there's no significant semantic difference between OP's exact two examples, but in certain specific contexts one or the other would be more "idiomatic".

1.) If you were sitting in a hotel having an early "working breakfast", you'd be more likely to use #1 (with the meaning *"Did you like getting up early this morning?").

2.) The hotel manager wanting to know if his guest would like an early or a late breakfast the next day would be more likely to use #2 (with the meaning *"Will it suit you to get up early tomorrow morning?").

Thus, although I don't see any distinction based on "repetition", there is a distinction based on the fact that the gerund/present continuous ("Do you like doing this?") is a better choice when the person you're asking is currently doing whatever it is.

The difference becomes clearer if we change the activity from getting up early to being here...

3: Do you like being here? (normal English)
4: ?Do you like to be here? (unlikely to be said by native speakers)

Where #4 is unlikely because (as explained above) we normally use the first form when the activity being asked about is currently being performed (and usually we'd only be asking such a question if the person was in fact "here" at the time). But contrast that pair with...

5: Do you like being at home on Thanksgiving?
6: Do you like to be at home on Thanksgiving?

In this case, #5 is more likely if you're asking someone who's currently at home, on Thanksgiving Day (or very close to it). Effectively, you're asking if the person likes their current circumstances.

A good trick to avoid such confusion is to remember that there are two different entries used to in a dictionary. The typical usages are:

• used [infinitive] is for habits in the past. Here, used to is a verb, e.g., I used to play badminton.
• used to [verb-ing] (also used to [noun]) is for familiarity. Here used to is an adjective, and you always need verb to be in front of it, e.g., I've lived in Central London for six years now, so I'm used to driving in London now. I'm used to the noise too.

Compare these two sentences:

I am used to playing badminton.
I used to play badminton.

The first (I am used to playing) means: I have played badminton often enough, so you can expect me to know what badminton is and to play it well enough. The second (I used to play) means: I played badminton regularly for a certain amount of time in the past. (I don't play it anymore.)

In your question, it would be ungrammatical if you replaced thinking with think. Do not write "He is used to think ...". Always write "He is used to thinking ..." to mean that he is familiar with that kind of thinking. However, if you want to mean that he changed his opinion, use "He used to think ...". For example, "He used to think smoking is good."