I always lean toward AMI, but before I tell you why I want to add that unless the school that you are considering is actually accredited by AMI you can only judge it by how it makes you feel.
Just because a teacher trained with AMI doesn't mean that she/he is following those standards.
As far as the suggestion that AMI is more "strict," what does this mean? When I was searching for a training center to take my first training (primary - 2 - 6+ years) I looked all over the country. I was just getting out of graduate school and nothing stood in my way to pursuing what I wanted to do which was to get the best education in Montessori principals available. I looked into every type of training and ultimately decided to go with an AMI center in MN. It was the best decision I ever made in regards to my education. I also then pursued an AMI elementary (6 - 12 years) diploma.
AMI teachers spend far more time learning the history of the method and the reasons behind everything we do than other trainings. I know this because I asked around when I was searching for a training center. We write our own "albums," which are the books where we write up the basic presentations we offer the children. Most other trainings, perhaps all of them, offer these albums for sale. We must write our own as an exercise in learning all of the presentations and materials.
While some site the longer "internship" that some trainings offer, I believe that while our internships may be shorter, we spend so much time doing observations and understanding what we are doing and why we are doing it that in the end it works to our advantage. A longer internship could potentially mean that a student spends a long time, the majority of the training time, learning from someone other than a trained teacher trainer and can learn some bad habits.
I had a good friend in training who had taken AMS training. She decided to go back to take AMI training and afterwards said that what she had thought would be the case was true - the AMI training was superior.
There are many things about some AMS schools that I find baffling, such as the lack of the three hour work period - something Montessori wrote about at length and which Montessori teachers have observed throughout the world for years. The three hours of uninterrupted work was fundamental in providing the child enough time to engage in a full cycle of activity.
At the elementary level at some AMS schools that I have known the use of check lists, the lack of "great work" and of a "going out" program are some of the things that I find sad. The level of enjoyment and fulfillment of work for the children is just not the same as when they can follow their own schedule, pursue and interest for hours or days and plan their own outings, even if only for one or two people, to satisfy a curiosity.
Is this the "too strict?" That the children actually have more freedom in an AMI setting? I find that ironic.
At am AMS school with which I am familiar (a close friend works there) all of the elementary students and even the "extended day" (the five year olds who stay for the afternoon) students have check lists. They have a quota of work they must do each day, especially in math and language. This friend says that there is no "great work" going on. Great work is when a child sets a project for themselves just because it interests them and works with intense concentration for many hours or days. An example from my own days in an elementary classroom include such things as learning all of the countries, capitals, rivers, and other major geographical features of Asia. Another example is building a pond/wetland which includes native plants and animals. One of these took several days, one took several weeks. The children devoted most of their time to these projects. For the Asia one, the child did no other work for about three days. At the end, he and his friend did know all that they had set out to know and were very pleased with themselves. They then moved on to other things. For the pond project, they spend most of the day for several weeks doing this project and worked intermittently in other areas. Most of their math was figuring out square footage of pond liner, volume of the pond, and calculating costs of the materials and plants. This cannot happen at my friend's school because the daily "work" required does two things: it interrupts any "great work" getting underway and it puts a "cap" on the work by making the child work to the level of a teacher's expectations rather than their own. Children left to their own, if they have been in this setting from an early age, almost always work far beyond what any teacher could or would require.
Are check lists ever appropriate? I would say probably not standard ones prepared by an adult. In AMI training we are taught that it is the goal to have all of the students working from their own heart. We view any "check list" as a temporary solution. The check list should be done *with* the child, either every morning or every couple of days, depending on the student. There should never be more than a couple of children doing this. If there are, then look at your teaching in general. There is something wrong. The child helps to come up with the goals, or perhaps comes up with all of them and the teacher just helps to guide. Again, this is a temporary solution for a child who is having a challenging time organizing their time.
Having said all of this, I know that there are some excellent AMS teachers out there who have done much studying and who naturally have a wonderful understanding of what Montessori is all about. At the same time there are some terrible AMI teachers who just don't 'get it.' You have to go with your experience of each setting.
Anyway, this is my take. I feel that I have a good perspective having explored both options and having a lot of exposure to both.
Good luck with your decision, and remember, teachers and schools do vary despite their training.