Learn English – Proper comma usage for multiple city/country pairs

Yes, this is correct. This is an example of parenthetical commas. You may also use em-dashes or parentheses here:

…the people that I work with, including clients, which I believe…

…the people that I work with–including clients–which I believe…

…the people that I work with (including clients) which I believe…

In this case, I find commas preferable.

First of all, punctuation is a matter of style, and you will find the rules for that style in the style guide that your employer has adopted. Which guide governs your edits, and what does it say? Different guides have different rules, but the good ones will emphasize that fiats must be tempered by the recognition of exceptions and the role of the good judgment of authors and editors.

That said, on to your specific questions. Your sentence has one clause with a compound predicate, which (as you noted) is played well and scored tons. Yes, Baxter is the subject of both verbs, but that doesn't make

scored tons of points

a clause on its own. The fact that each verb governs a different object doesn't matter. You could have

Baxter played his role well in the first game and so continued throughout the series.

The second verb (continued) doesn't have a direct object at all, and you still have the comma dilemma.

I use the Chicago Manual of Style, which recommends that commas separate conjoined independent clauses and that they do not separate compound predicates. In fact, CMOS states that these rules are more than recommendations, calling them "obligatory", but CMOS notes at least one exception to the former -- short clauses. Their example that omits the "obligatory" comma is

Charles played the guitar and Betty sang.

That exception won't help you with your problem, though. You should probably consider inserting theforbidden comma when not doing so will create a so-called garden path, an invitation to the reader to choose the wrong parse. In this case would the absence of a comma lead the reader to initially consider that the conjunction was to join something other than the second predicate? For example, a compound direct object, along the lines of

Baxter played his role well in the first game and the second.

or an adverbial phrase as in

Baxter played his role well in the first game and in general.

And here's where your judgment as an editor comes in. My personal opinion is no: the reader immediately encounters a verb (scored) and not an article (and the second) or a preposition (and in general). But there's a simple way to avoid the agony, and that's to make a simple edit to create a compound clause to justify the comma:

Baxter played his role well in the first game, and he scored tons of points throughout the series.