Your distinction between "true" and "logical value 1" is not one that formal logic generally observes. Here "1" and "true" are synonyms for the same concept.
The meaning of the $\Rightarrow$ connective is what its truth table says it is, neither more nor less -- the truth table defines the connective (in classical logic). Fancy words such as "implication" or "if ... then" are just mnemonics to help you remember what the truth table is, and what the connective is good for -- but when there's a conflict between your intuitive understanding of those words and the truth table, the truth table wins over the words.
The important thing to realize is that $\Rightarrow$ is designed to be used together with a $\forall$. If you try to understand its naked truth table it doesn't seem very motivated -- certainly it can't express any notions of cause and effect, because the truth values of $p$ and $q$ just are what they are in any given world. As long as we're only looking at one possible state of the world, there's not much intuitive meaning in asking "what if $p$ held?" because that implies a wish to consider a world where the truth value of $p$ were different.
The device of standard formal logic that allows us to speak about different worlds is quantifiers. What we want to say is something like
In every possible world where I put in a coin, the machine will spit out a soda.
(though that is a little simplified -- we want to consider a "possible world" to be one where I made a different decision about my coins, not to be one where the machine had inexplicably stopped working even though it does work now. But let's sweep that problem aside for now).
This is the same as saying
In every possible world period, it is true that either I don't put in a coin, or I get a soda.
which logically becomes, using the truth table
For all worlds $x$, the proposition (In world $x$ I put in a coin) $\Rightarrow$ (In world $x$ I get a soda) is true.
Since there's a quantification going on, the truth value of the whole thing is not spoiled by the fact that there are some possible worlds with a broken machine where the $\Rightarrow$ evaluates to true. What interests us is just whether the $\Rightarrow$ evaluates to true every time or not every time. As long as we're in the "not every time" context, the machine is broken, and that conclusion is not affected by the "spurious" local instances of $\Rightarrow$ evaluating to true in particular worlds.
The construction that models (more or less) our intuition about cause and effect (or "if ... then") is not really $\Rightarrow$, but the combination of $\forall\cdots\Rightarrow$.
Unfortunately in the usual style of mathematical prose it is often considered acceptable to leave the quantification implicit, but logically it is there nevertheless. (And to add insult to injury, many systems of formal logic will implicitly treat formulas with free variables as universally quantified too, so even there you get to be sloppy and not call attention to the fact that there's quantification going on.)
Note also that this is the case even in propositional logic where there are no explicit quantifiers at all. To claim that $P\to Q$ is logically valid is to say that in all valuations where $P$ is true, $Q$ will also be true -- there's a quantification built into the meta-logical concept of "logically valid".
The claim is not a claim of the form that is considered in propositional logic -- it belongs squarely in predicate logic.
Some authors of introductory texts go to the trouble of defining a quasi-formal concept of what the word "proposition" means. Usually not much is actually done with this concept and it is forgotten about completely when you get to define propositional logic formally. Often it looks like the main purpose of offering the definition is to attempt to explain why it's called "propositional" logic.
Unless you anticipate being asked "is such-and-such English sentence a proposition?" in an exam, I would not worry about a particular author's definition of the word. It is not going to be important.
Best Answer
The formula $$x=1$$ is variously called a predicate, a propositional function, and an open formula. You're right about this formula not being a proposition: a predicate typically has a varying truth value. (The predicate $x=x$ is definitely true though, and is thus a (logical) validity.)
Similarly, $$\text{if }x = 1,\text{ then }x + 1 = 5\tag{*}$$ is also a predicate. It has the opposite truth value as the first predicate above.
On the other hand, the formula $$\text{for each $x, \, \big($if }x = 1,\text{ then }x + 1 = 5\big)\tag{#}$$ is a (false) proposition, also called a sentence and a closed formula. As you've pointed out, (in each interpretation) every proposition has a definite truth value.
What distinguishes a predicate from a proposition is whether the formula contains any free variable, that is, whether each variable in it is quantified by either $\forall$ or $\exists.$ (As illustrated above, a formula that has a fixed truth value is not necessarily a proposition!)
In practice though, implications in mathematical writing are typically implicitly universally quantified. This is the real issue in the given example: the predicate $(*)$ is actually intended to be understood as the proposition $(\#).$