Learn English – ‘Provides me with an excuse” or “Provides me an excuse”

The question When should a verb be followed by a gerund instead of an infinitive? on English Language and Usage is what you need. The first answer contains lists, which I'll reproduce here

There is a copy of The Brief Penguin Handbook here; the Verbs Followed by Gerunds or Infinitives section is here.

The only way to know is to memorize which verbs are followed by gerunds and which are followed by infinitives. The relevant categories are:

Verbs Followed by Infinitives

Most verbs are followed by infinitives. If the verb is not found in the list below, it is probably followed by an infinitive.

Verbs Followed by Gerunds

The verbs in the following table all need to be followed by gerunds.

The students don’t enjoy going over the same rules again and again.

• admit (to)
• appreciate
• avoid
• be accustomed to
• be used to
• can’t help
• consider
• delay
• deny
• discuss
• dislike
• enjoy
• feel like
• finish
• get used to
• imagine
• keep (on)
• look forward to
• (not) mind
• miss
• postpone
• practice
• put off
• recommend
• regret
• report
• resent
• resist
• risk
• stop
• suggest
• tolerate
• understand

Verbs Followed by an Object Before the Infinitive Verb

I advise you to go to school early today.

• advise
• allow
• *ask
• authorize
• cause
• challenge
• convince
• encourage
• *expect
• force
• get
• help
• hire
• inform
• instruct
• invite
• *need
• order
• permit
• persuade
• *prepare
• *promise
• remind
• require
• teach
• tell
• train
• urge
• *want
• *warn
• *would like

*Some words can be used without an object as well as with an object.

I want him to go. I want to go.

Verbs Followed by Either Gerund or Infinitive

Sometimes the meaning changes according to the verb used.

He doesn’t remember giving the homework to Mr. Young.
He didn’t remember to give the homework to Mr. Young.

• attempt
• begin
• can (not) bear
• can (not) stand
• continue
• hate
• intend
• (dis)like
• love
• neglect
• prefer
• remember
• start
• try

Verbs Followed Only by the Simple Form (no “-ing” or “to”)

Four verbs are called causative verbs. They are followed by an object; the verb after the object is always in the simple form:

• let: They let him go on the trip (instead of “let him to go”).

I let him take my book home for one night.

• make: We made her do her chores first.

I made my sister cry.

• help: She helped her finish her homework.

I helped him find the bookstore.

• have: The teacher had him stay after school.

I had my teacher explain the answers.

Verbs Followed by Either the Simple Form or the Gerund (no “to”)

Some verbs are called verbs of perception and are followed by either the simple form or the “-ing” form.

• see: I see him go.

I see him going.

• notice: I notice him run to school every day.

I notice him running to school every day.

• watch: I watch him struggle with his homework.

I watch him struggling with his homework.

• hear: I hear him sing.

I hear him singing.

Others including: look at, observe, listen to, feel, smell.

The initial page actually has anchor links for each word, making it easy to see what the web page is trying to convey.

With an intransitive verb, objects and complements are included in the predicate. (The glacier is melting.)

The objects word links to the Direct and Indirect Objects and thus refers to either or both direct objects and indirect objects. This is further a nod to both intransitive verbs, which don't require either, and ditransitive verbs, which can take both at once.

The complements word links to the Complements section and thus refers to words or phrases that complete the sense of a subject, object, or verb.

So, let's bring that back to the sentence and its accompanying example:

• Not only is the glacier the subject in action (the glacier melts), but also receives said action (the glacier is being melted as a result of its melting).
• The word melting is a complement that completes the sense of the verb is here, and thus is a verb complement.

In your defense, the page isn't terribly well organized and, as Yuri points out in her comment, suffers from inconsistent nomenclature. I'm pretty sure that you're neither the first nor the last person to be confused by that web page (yeah, it was confusing me for a while too).