You have become a bit confused about the statement in the link that you have provided. What it is saying is that the inflections on the end of the verb are the same as those of the "bare infinitive". In other words, there aren't any! It isn't making any other claims about the similarity between the bare infinitive and the subjunctive. In fact, they are completely separate moods (the moods are indicative, infinitive, imperative, and subjunctive, in my best guess as to their frequency of use) which each have their place in the language.
So, the link is simply saying this: for the subjunctive, instead of saying "he works" you say "he work". That's the only difference, except in the verb "to be". In that case, the present is the same as the bare infinitive "be": I be, you be, he be, we be, you be, they be. Also, to be is the only verb where the past subjunctive is different from the past indicative: I/you/he/we/you/they were. Have a look at these:
It is necessary that he work on Saturdays.
I would have preferred that he worked only on Saturdays last summer.
The first is an example of the present subjunctive, the second of the past subjunctive.
So, to answer your questions. First question: if you see a bare infinitive that isn't complying with the rules you describe, it's probably because it's a bare infinitive! Consider these examples:
I make him work on Saturdays.
I made him work every Saturday last month.
These are examples of the use of the bare infinitive. Have a look at this for more examples of the use of bare infinitive. (By the way, I prefer to think of "Let's play Monopoly", which they give as an example of the use of bare infinitive, as the first person plural imperative, but I won't go so far as to say that it's wrong to call it a bare infinitive.)
Second: no, a bare infinitive is entirely distinct from a present subjunctive, except in the sense that I described in the first paragraph. This might help: the subjunctive mood is inflected like the bare infinitive (there are no inflections, or alterations to the ending of the verb), but constructed like the indicative mood (I/you/some other pronoun or noun plus the verb).