IMO one of the essential aspects of improvisation is flow. It is especially true for those that have spent years working on the foundational aspects of learning to play the instrument, with very little to no improvisational activity. From my own experience being trained as a classical pianist, there was much emphasis on the details of making the music sound "proper." Indeed there is much to learn in the craft of playing classical music, but I also think if left unchecked that this proper culture can have a negative impact on a musician's ability to tap into creative flow. A lot of the study workflow in classical training revolves around taking something, scrutinizing it, and refining it a little at a time. Rinse and repeat. Improvisation can at times seem at odds with such a workflow, for a person that has become accustomed to a certain way of working with music and ideas.
In order to achieve flow, one needs to let go of all those rules. Let go of the need to play "properly." For someone that is new to improvisation or reluctant to improvise, achieving flow is more important than following rules. One has to be able to give oneself the permission to become vulnerable. As one's ability in flow develops, guidelines can be gradually introduced to provide direction. IMO rules in improvisation are not about analyzing what is right and wrong, they are tools to give direction towards things such as: style, mood, and evoking specific ideas.
As for an exercise at the instrument: start with a simple single idea - a note or a chord. Repeat it, again and again. When your sense of flow tells you something needs to change, add another note to the first idea. Don't think about rules or choosing the theoretically correct note. Be aware of the sound of your instrument, it will tell you what to do next. Be aware of your sense of flow, it will tell you when and how to move. Continue in this manner creating an improvisation by adding and subtracting ideas from the current one being played.
You might find it helpful working with an improvisation mentor to do some guided improvising. Sometimes one tends to default to familiar techniques when in unfamiliar territory, and the gentle guidance of a teacher can encourage a person to retain focus on an improvisation mindset.