I believe there are different things we use the term 'equation' to mean over the course of education, and students can get confused because it's usually never explained where the distinctions come in.
At a simple level, an equation is any statement that two things are equal. It is a logical statement with a true/false value that uses 'equals' as the relation, rather than, say, less-than. It's worth remembering that 'equals' means different things in different contexts.
So then different things can happen. One option is that the logical statement is false, e.g. $2+1=5$. We don't use these that much in education, so students may well get confused unless it is made clear that they are being taught to evaluate rather than being taught facts.
Another option is that the statement is true, e.g. $2+1=3$ or $2x+x=3x$. We start teaching simpler forms of these at primary school, and since this is pretty much the only form around, 'equation' does the job, if needed at all.
The third option is that we have decided to assume that the equation is true by hypothesis, and wish to determine the consequences of that hypothesis, e.g. $2x+1=3$. We introduce these later as 'algebra'. Here students will see both cases: $2x+x=3x=5$. These are generally simple enough that they can distinguish between 'true' and 'true by hypothesis' without realising that there's a distinction. In this context we tend to use 'solve' to mean the action of determining the value(s) of the variable. But we also use 'solve' for questions that aren't obviously of this form, so students may not pick up the technical meaning.
After a bit, we start to introduce other statements like $\sin^2(x)+\cos^2(x)=1$, which are of the always-true type, but are not obviously so. We call these 'identities'. These are, in a sense, equations that 'cannot be solved' (more accurately, solving them would not yield much useful information about the value of the variable). Some students will not pick up this distinction, so 'identity' becomes a word they don't understand but think they should (and therefore possibly scary).
Along the way, we also introduce 'equations' with other relations, e.g. $2x\leq 3$. Since we typically use only a limited range of relations, and may not supply an alternative term, these might also get categorised as 'equations'. After all, basically the same methods are used to 'solve' them.
In addition, we bring in questions where the solution requires a sequence of logical statements rather than simply a sequence of equal expressions. Again, students may not notice the change, and come to think of '$=$' as meaning 'the next bit of the solution'. So 'equation' is no longer 'a statement with $=$'.
So then they reach university, and start getting picked up on these distinctions. Maybe they then start to catch on. Meanwhile, they meet definitions. And after a bit one might ask 'what is the definition of 'equation'?'.
Perhaps, then, the answer is partly that there isn't a single definition, because 'equation' is both a mathematical term but also a meta-mathematical term. That is, we use it to talk about doing maths, as well as about the content of maths. And education terms aren't that well suited to rigorous definitions.