Question: A friend recently told me that "can" is a rare verb without an infinitive. I have since looked it up and discovered it is an auxiliary verb. In my mind it modifies a "proper" verb in much the same way an adverb does.

"I *can* **jump** puddles"

"I *could* **do** it"

"to jump" and "to do" are OK, "to can" is wrong (meaning "to be able to").

"I *am able to* **jump** puddles"

"I *was able to* **do** it"

So, are words like "can/could", "must", "will/would", etc. actually verbs or are they only auxiliary or modal verbs and not "proper" verbs? I was also chatting to a Frenchman and he said in French auxiliary verbs are just called auxiliaries and are not considered to be verbs, so is it the same in English? It seems to me that if you can't "to" or "-ing" a word it is not proper verb.

"to jump", "jumping" - OK

"to must", "musting" - wrong

However, if auxiliary and modal verbs are actually verbs is there a word to describe a non-modal, non-auxiliary verb?

Please, no comments about canning factories or making wills. I already know that the words "can" and "will" can be infinitives but with a different meaning.

**EDIT**

Black bears, brown bears and polar bears are actually bears. Koalas and pandas are not bears yet some people call them koala bears and panda bears because they look a bit like bears. Likewise jellyfish and starfish are not fish but are aquatic animals as fish are. Alternative names for them are sea jellies and sea stars. Ironically, the seahorse is actually a fish. Things are not always what they're called. So are auxiliary verbs actually verbs or just called verbs?

Answer: 
If you are building a parser, you will separate modals, auxiliaries, and lexical verbs; mixing them up would be ungrammatical. You can lump them together as verbals (components of predicators), and equate that group with the historical verbs (named as a part-of-speech back when it was important not to over-complicate the analysis).

P.S. Beware the sentence fragment. Having a modal without another verbal does not force it to be a verb.

P.P.S. Modals could be stored like adverbs, or they could be stored as part of the auxiliary (components of memory are typically elided in discourse if they can be assumed); but 'adverb' is already an overloaded part-of-speech.
There are different views on the things you ask about and the major grammar texts do seem to differ in the details, so there are "traditional grammars" and "modern grammars" as authors grapple with various complexities and redefinitions and have different views.

I'm not sure about what you call a "proper verb" (a judgement call really) but the term 'main verb' is similar to what I think you mean.

Most simply:

A "simple verb" consists of a single "verb" (a 'main verb') in the "verb phrase". 

A "compound verb" consists of more than one verb in the verb phrase. 

These extra verbs are "auxiliary verbs", this later group consisting of the "**primary auxilaries**" ('have', 'be', 'do') and the "**modal auxilaries**" (a closed group of 9 including 'have', 'can', 'will'). [This is one classification system of verbs, though here not including copular verbs]

The auxilary verbs bring something extra to the verb phrase (for example assisting with indicating tense, mood, modality). 

Modal auxilaries (like 'can') are defective in that they do not have all the properties of a 'main verb': they have no infinitive form, no regular adjustment for tense (excepting a historical use of shall/should for example), and only a single form for all persons. There are also the first verb in a verb phrase. They do not have past or present participle forms, so 'musting' is ungrammatical. 

The primary auxilaries 'have' and 'be' can also function as main verbs. I think that 'can' can too but only when it not used as a modal auxilary, i.e. when it has a certain different meaning. One can provide the following advice to their teenage children playing music too loud in their room: "Can the noise now please!" (where 'can' means 'turn it off'), but this is like your note about canning factories - its a different meaning to say " 'to can' the beans" - from what you were asking about.