Is "to" really part of the infinitive?

Consider this:

I like to eat here.

vs

I would eat here.

It appears to me that "to" has nothing to do with the infinitive form of the verb that follows. It is, in this example, an integral part of like to, not of to eat.

Is my thinking flawed? If you think it is, could you please explain which form of the verb "eat" is used in the second example, if not the infinitive?

It is probably a matter of definition, not of true inherent "belonging to". I believe the etymology of this kind of "to" is the usage of a preposition before the infinitive, in order to indicate a relation of direction or purpose between a finite verb and the infinitive: I went to school; I went to pick him up. This probably originated in predicates with verbs that have a direction. But that was long ago, when the infinitive still had a distinctive form in English or Proto-Germanic; I think it was something like *eatan (don't pin me down on this, I haven't looked it up). Later, when the infinitive became indistinguishable from other forms of the verb, this to evolved into a more general marker of the infinitive. So the most precise definition would probably be to say that to belongs neither to the finite verb ("like") nor to the infinitive.

That said, you could say "like often goes with to, so we say to belongs to like", or "to often introduces or links to infinitives, so we say it belongs to the infinitive". In lists of phrases, educational books often tend to emphasise the connection with like; in passages that describe the infinitive, they tend to treat it as part of the "full" infinitive, as opposed to the "bare" infinitive in "I will go". I believe this is the most traditional way to describe it.

Here are the two main bits of positive evidence that to is part of the infinitive.

1. It doesn't show up with nouns in the same situations.

• I like to dance.
• I like pizza.

Notice that to is not an inherent part of like, because even in the exact same sense of the word, if it is followed by a noun there is no to.

2. It shows up without another verb.

In phrases without a verb in front of the to, you can see it appearing:

• To live is to dance.

For modals, as others have explained, the modal verbs do something special with the verb, as do the helper verbs "to be" and "to have". But they are a special case. Modal verbs also don't allow a gerund after them in any situation, e.g., you can't say "*I would walking" like you can with a verb that is usually followed by to. But with "like" you can say "I like to dance" and "I like dancing". Only the form of the verb is changing, from the infinitive to the gerund.

The second sentence does use the infinitive, but would is a modal verb, so it kicks out the to. Wikipedia explains:

The main verb that is modified by the modal verb is in the infinitive form and is not preceded by the word to (German: zu, Dutch: te). There are verbs that may seem somewhat similar in meaning to modal verbs (e.g. like, want), but the construction with such verbs would be different

Note how you wouldn't say "*I can to eat here" or "*I must to eat here", either.

No, it is not. This is probably one of the biggest misconceptions in all of English grammar/linguistics.

The bare infinitive IS the infinitive.

It's better to say that "to" accompanies the infinitive depending on how it is used, but it's not part of the infinitive nor does it complete the infinitive.

When the infinitive is used as an adjunct or a complement of another word, it cannot be used alone. It's needs the grammatical particle in between to "bind it", so to speak, to the word that it's complementing. This has parallels in many other languages:

something to eat

quelque chose à manger.

algo para comer

etwas zu essen

In the example you gave above, you are correct. To, in that context, is part of the phrase "like to", which is an auxiliary with the same grammatical function as 'would'. In compound verbs with modal auxiliaries the bare infinitive is used.